How do football players or teams decide on the type of facemask on their helmets?

I don’t know about the situation in football, but hockey players are free to choose their own type of helmet. As long as it fits regulations, they decide whether they want to wear a visor (and whether it’s tinted) or even a full face guard, which some players do after a face injury. Goalies are similar – they have to have some kind of face guard, but whether it’s plexi or bars is up to them.

After a little research, this is what I found on Wikipedia.
Also, a more recent addition to the football helmet is the visor or eyeshield, which is affixed to the facemask to protect players from glare or eye injuries, such as pokes. [...] High-school and pee-wee leagues prohibit all but clear visors. This rule was enacted so that training staff and coaches can easily view a player’s face and eyes in the case of a serious injury, to discern if the player is conscious. The NCAA banned the use of tinted visors for the same reason, and the NFL has followed suit as well. However, players with eye problems may still obtain special permission to wear tinted visors, some notable examples being LaDanian Tomlinson and Chris Canty.
The one-bar facemask has been illegal in the National Football League since 2004, but a grandfather clause allows players who wore the mask prior to 2004 to continue to do so for the remainder of their careers. Free agent punter Scott Player currently wears a 1-bar facemask and will be the last unless a new rule reissuing the 1-bar facemasks is issued and passed. Since Scott Player is approaching 40 years old and did not play at all in the 2008 season it is unlikely that the 1-bar facemask will be seen in the NFL play again. Player currently plays for the UFL’s Hartford Colonials—the 1-bar facemask is legal in that league.

I don’t know for sure, but it sounds like the Equipment Manager.
They are the ones who also decide what kind of shoes the players wear, depending on field conditions.

It has to do with the position they play. Obviously, with more coverage of the face, chances of facial injuries go down exponentially but, all those bars can cause problems with visibility and blind spots when looking for players/the ball at 30+ yards away. Players who don’t need make a play on the ball, but on the player instead tend to wear heavy guarding facemasks whereas players like quarterbacks and wide receivers need to be able to see a large portion of the field and defenders on it all at once. Players are free to chose their type of facemask but coaches and staff are there to make sure the players are protected and as resistant as possible to injuries but also that they can perform their job.

I have seen high school corners and running backs wearing full linemen facemasks because it was what their mom bought them and wouldn’t spend the money on the “right” mask or because they were trying especially hard to avoid injury.

The league sets some of those guidelines.