As you proliferate, you can choose any players that have counters, including yourself.

You can choose any permanent that has a counter, including ones controlled by opponents. You can't choose cards in any zone other than the battlefield, even if they have counters on them, such as suspended cards or a Lightning Storm on the stack.

You don't have to choose every permanent or player that has a counter; you choose only the ones you want to add another counter to. Since "any number" includes zero, you don't have to choose any permanents at all, and you don't have to choose any players at all.

If a player or permanent chosen this way has multiple kinds of counters, only a single new counter is given. You choose which kind of counter to give that permanent or player as you proliferate.

Players can respond to the spell or ability whose effect includes proliferating. Once that spell or ability starts to resolve, however, and its controller chooses which permanents and players will get new counters, it's too late for anyone to respond.

If a creature has +1/+1 counters and -1/-1 counters on it, state-based actions remove the same number of each so that it has only one of those kinds of counters on it. There won't be any counters of the other kind to put onto it as you proliferate.