One of the worst feelings in the world as a GM is when you recognize that there is a player sitting at the table not being engaged by the game. It can really throw you off your rhythm. Are they not having fun? Is the game not going the way they expected? Do they feel useless? Powerless? Did something bad happen? The questions can get pretty endless, and all of them point to an easy solution: you are a bad GM.
This is matched by the fact that as a player - and we've all been there at least once - being the player sitting back, quietly, and not getting any screen time can be boring. I've often said that one of the most boring things I can think of is watching a group you're not a part of play a RPG. My view on that - for some groups (such as critical roles) - has changed over the years, but by and large if you're not engaged by the game on some level you're basically just sitting in a chair of questionable comfort levels listening to other people have a good time.
So, when this happens, who is at fault?
The GM's Responsibility
As a GM you have the responsibility to run the game. This game should be the one that you want to run, but it should also be the one that the players want to play. There is no point in only the GM - or only the players - being invested in a game because it makes the other group miserable.
As the GM your job is to provide scenarios, settings, and NPCs to interact, prompt, poke, jar, and otherwise get the PCs into action. If the PCs are pro-active, then your job is to have those things react to the PCs and keep the ball rolling. Ideally, you should have things for each of your player characters to do and challenges for each of your player characters to overcome. However, while your job is to provide the hook for things to do, it is not your job to make the player engage with the hook or to pull them into the game. That is their job.
The Player's Responsibility
As a player you have the responsibility to generate a player character that is capable of being the protagonist of a story with an arc for growth. However, this character needs to fit into the game that the GM and your fellow players have said you were going to run. If the GM wants to run a game about everyone being super heroes who save the day and help people, you shouldn't be dead set on playing a super villain, an anti-hero who doesn't care about collateral damage, or the nosy reporter looking to expose all the super hero identities. Not that those characters can't work, but they run counter to the purpose of the game.
Your other responsibility as a player is to get yourself into the game. The GM - who most people think is responsible for getting people into the game - has three to five (or more) other players to deal with, plus the world, plus all the rules, plus all the NPCs. You, on the other hand, have one thing to deal with: your character. The GM should provide a situation to get involved with, but you should be the one to make your character grab onto the adventure hook and see where it takes you.
Where Is The Failure
So whose responsibility is it for an unegaged PC? Well, that depends on where the failure happens. Is the hook there but the PC doesn't grab it? This I tend to put on the player most of the time. Why? Because in most situations I have seen a hook not appealing to a PC means that the PC isn't made for the kind of game the GM discussed with the group and people made characters for. To go back with the super hero game, if a super villain starts robbing the city mall while your character is there and your response is to go home and watch it on the news in relative safety...you didn't make a character for a super hero game.
The time where it falls on the GM is when the hook isn't designed for the kind of game that was discussed. A hook of helping an old lady with muggers is great for a super hero game, but less capable in a game of super mercenary shadowrunners who don't do anything without a paycheck on the other side. An offer to join the King's army and go to war against the enemy country of Nefrandel isn't going to broadly appeal in a game about looting ancient tombs and lich towers for fame and fortune.
There are other problem points as well. Player A may latch onto a hook meant for Player B without realizing - or even while realizing - and that makes Player B feel redundant and useless. Player C may have made a character that does everything Player D does, only better - or in addition to other stuff. The game may have drifted from its original intention, and now the only character who still fits the original idea is being left behind with the new characters and new adventures.
Ultimately though, if you're a GM all you should need to focus on is providing something to do.
As a player, you should focus on getting your character involved.
Let the dice handle the rest.