The question of "Who has the final say?" was brought up on The Herd in 2017. I know 'cause I copied the post and accompanying links for future reference. I figured it might come up again. Anyhow, I'm not sure how much of this still holds true because although Sneed and Demoff are still there, Fisher is not, I'm sure some of this is still relevant, though.
The following info is from a 2/15/2012 Jim Thomas article in the St Louis Post-Dispatch:“This is a three-man team now, and we’re looking forward to it,” Fisher said. “There’s no doubt in anybody’s mind that this is a partnership, and we’re gonna move forward. We’re going to make the right decisions; we’re going to make the correct decisions in all aspects of this roster.”
In the partnership’s most common form, Snead will identify and evaluate prospective talent, Demoff will sign it, and Fisher will coach it. But when it comes time to pull the trigger on a free agent, or turn in the card on draft day, where does the buck stop? Who has final say?
“We don’t anticipate differences of opinion,” Fisher said. “But in the event that there are, we will move on to the next player. It will be a consensus. As we said, it’s a partnership.”
If the Fisher-Snead-Demoff troika can’t agree on whether to take a particular player, they just won’t take him. However, Fisher said he doesn’t see it ever being that way on draft day. Any arguments or debate will take place well before draft day.
“The work will be done prior to the draft,” Fisher said. “The time that we’ll be spending collectively together is enormous right now. And I would say a week before we’re ready to pick, we will have made all our decisions.”
The info below is from a Feb 2012 article by Bernie Miklasz in the St Louis Post-Dispatch:Question: * Does Snead work for head coach Jeff Fisher, or is Fisher working for Snead? When Snead made a reference to working “for” Fisher it set off some clucking among those who are hung up on titles.
Here’s your answer: Fisher and Snead work for team owner Stan Kroenke. Fisher didn’t ask for total roster control as part of his contract with the Rams, but he does want to have a say, and he expects to have influence, and I don’t think anyone at Rams Park will push the coach out of the way when it comes time to make big decisions. But Fisher is a football coach. He doesn’t have time to scout and grind tape. Fisher wanted to bring in someone who could oversee the vitally important scouting work and then help the Rams make smart personnel decisions in the draft and free agency. And Fisher wanted someone he can trust. Snead is a workaholic that wears out DVD players with long hours spent studying video of players. And in Atlanta he had an eye for talent.
In April 2016 there was an article "Who calls the shots in all 32 NFL draft rooms?" on ESPN.com. Here's the answer given by Nick Wagoner for the LA Rams:
Los Angeles Rams: Like many teams, the Rams prefer to say that all of their picks are a group consensus, but if it really comes down to it, coach Jeff Fisher has the final say. The Rams gave Fisher a five-year deal worth around $7 million a season in 2012, and with it, they gave him a lot of say in personnel matters. He was already in place when the team hired general manager Les Snead, and the two generally work together on building the roster. -- Nick Wagoner
I can't say how it is today with McVay, Demoff, and Snead. But this is how it was with Fisher, Demoff, and Snead.