Well, first off, I'll be keeping a watchful eye on this thread. Keep it civil and don't give me extra work to do. >:[
And now my thoughts. I haven't yet read any of the other responses, so don't be shocked if others have already said some of this.
should those officers be tried for the shooting of Alton Stirling and Philando Castille?
Perhaps. Trials are not the alpha and omega of dealing with potentially unlawful actions. First of all it should be investigated and the local district attorney's office needs to decide if it looks like a crime rather than police lawfully using deadly force, then if the DA calls it a possible crime charges need to be brought against them and the info needs to be brought before a grand jury (assuming these states operate on that same system, I don't want to go look the specific info up right now) for them to decide if there's enough to bring it to trial, and then if the grand jury says yes it should go to trial. The officers involved should be put through this process, and should either case make it to trial then alright, go for it.
Now, since you're going for personal opinion here rather than an explanation of the justice system, I think the legal severity of the Alton Sterling case hinges entirely on whether or not he actually had a gun on his person. I haven't seen it reported one way or the other, just reports that the police and DA's office refused to answer that question and others about the investigation. If he had a gun then the videos that have been put out (with their lack of visuals on Sterling's hands but full audio) make perfect sense as police using justifiable lethal force: they were called out to check on a report of someone threatening a homeless guy with a gun, Sterling apparently matched the description, they got into an altercation, once he was on the ground one cop yelled what sounds like "he's going to his pocket he's got a gun" and then he or the other one gets his gun out and yells "hey bro if you fucking move I swear to God" and then comes a yell of "??? he's going for the ???" (question marks for words I couldn't make out at all) and shots are fired. This sounds like they saw what appeared to be a gun, Sterling was reaching for it, one of the officers warned him not to move, he kept going for the gun, so the cop with reasonable fear for the safety of himself and his partner opened fire. If he did NOT have a gun though, then things get tricky: all you need for justifiable use of force is the
reasonable belief that your life is in danger to the point that you require lethal force to protect yourself. This fatal shooting can hypothetically be justified based on the circumstances even if Alton Sterling had no gun, just based on him having something in his pocket and him reaching for it. I think the most that these officers could get in trouble for is excessive force, perhaps for the start of the altercation (no video released so far shows why the cops took him to the ground) or for the four shots fired after the first two had presumably incapacitated Sterling. I don't think they'll get tagged for murder because it can't be proved beyond a reasonable doubt unless the store security camera footage shows a very different story than those cell phone camera videos.
The Philando Castille case is in a very similar boat, though as far as I know the only footage of the incident that exists is from the driver, Castille's girlfriend, who started filming after the shooting. The video itself contains everything you need to know to see why this case is very likely not going to end in a murder conviction: the cop in the video yells "I told him not to reach for it, I told him to get his hand off it." This was after the driver said that Castille had informed the officer that he was licensed to carry a firearm and was reaching for his wallet. Here's the situation: Castille told the officer he has a gun, he reached for his pocket, the cop told him to stop, Castille apparently did not stop, so the cop was afraid that Castille was going to pull a gun and fired on the uncooperative and armed man. And yes, the fact that he did not stop when ordered, even though he was reaching for his wallet to get requested identification, is considered being uncooperative. We don't know if Castille or his girlfriend were being confrontational with the cop before this, we don't know what the tone of the discussion had been, we don't know anything at this point except the aftermath... in which the cop seems extremely distressed by the fact that he just shot and killed a man. That sort of thing isn't exactly legally solid, but you're gonna have a hard time convincing a grand jury that that cop was some ruthless murderer after they hear how he was losing it after the incident. I don't think this one will even get to an indictment, honestly. People may not like it, but the laws for self defense and justifiable police homicide are far looser than "it's only okay if the person was actively trying to murder you," and this seems to fit right in the justifiable use of force category, as unfortunate and entirely preventable as it was.
Do you think officers should use deadly force?
Yes. Without the threat of deadly force, how exactly could they deal with someone who has a gun? Law-abiding citizens and criminals alike have ready access to firearms, and police need to be able to respond in kind with that on-demand deadly force to be able to do their jobs. What people need to understand is that the vast majority of cases in which people are killed by police are when the suspect very obviously has a deadly weapon. They aren't perfect, and sometimes you get these cases where someone either has a weapon or appears to have a weapon and then shit gets out of hand and someone gets shot, but frankly speaking 9 times out of 10 that is the fault of the suspect. Sterling could have stopped struggling and very likely avoided being shot, Castille could have stopped reaching for his wallet when the cop told him to stop and he almost certainly would not have been shot (because the cop would have asked him to step out of the vehicle to retrieve the gun and and identification himself, thus no need for the panic of self defense). It really sucks when the police kill someone that wasn't actively and blatantly being a threat to their person, but something being sucky doesn't mean it is or should be illegal. The fact of the matter is that about 1 in 3 people in the US own a gun, so police have to take that potential threat extremely seriously lest they end up dead themselves.
Should the Police dept in that community do a better job of building relations with said community?
Police having good relations with the communities they operate in is always a good idea. Whether or not they should do a "better" job depends on how good a job they're already doing. All that's needed is a general feeling of mutual respect, not being super friendly with everyone. So long as the people respect the police who are working to protect them and their belongings, and the cops respect the people that they are working to protect, then that's all you need. There need be no love going either way, just civility and understanding.
What about BLM? Are they right? Are they misguided? Do their complaints hold merit?
That there is a complicated set of questions. They're right that some places have problems where minorities are mistreated by the police. Complaints revolving around unjustified police actions against minorities at a very disproportionate level compared to such actions against white people hold plenty of merit.
Where they lose me is when they start saying shit about how
all cops are racist, or
all police departments are racist, or similar stupid generalizations. Hell, even saying that the justice system is racist because black people are disproportionately represented in arrests and convictions isn't actually a clearly merit-filled complaint. Black people do make up a disproportionate amount of prisoners and such, yes, but
they also make up a disproportionate amount of the people committing crimes. There are numerous socioeconomic theories as to why this is the case, and the cycle of shit where black folks being sent to prison a first time makes it far more likely that they will be lifetime criminals since having a record cuts off a lot of their limited opportunities, but putting it plainly it's not the job of the police to try to solve social ills. Their job is simple: try to prevent crimes from being committed, find and catch suspects if a crime is committed, investigate to prove who committed crimes. It's a self-propelling cycle where crime = punishment = more crime = more punishment and so on, but it is not in fact a one-sided issue of the cops just being racist assholes. It is an indisputable statistical fact that black people in the US commit a disproportionate amount of crime, therefore it makes all the sense in the world that they represent a disproportionate amount of convicted criminals. It's shitty, and it's a lot more complex than simple-minded bullshit like "cops are just racists" or "black people are just criminals" (which are the two stupid positions you see from the extremes, BLM being part of one of them), so just blaming the justice system is very misguided. Their methods are also pretty damned misguided, since all they're really accomplishing is sowing more discord. Y'know, like when BLM demonstrations start up chants like "more dead cops." That is not a good method to try to enact positive change.
So some of their complaints may have merit, some of their positions on things may be right, but I guess I have to firmly answer yes to the 'are they misguided?' bit. They're taking a very complicated issue and trying to boil it down to a few simple answers, and whenever someone does that it inevitably turns into a misguided shitshow. This one seems to be turning into an especially large shitshow.