In correspondence chess the button says Request a rematch. I like offer better.
Who should offer rematch?

I think both players may ask; analizing and considering the meanings of the words, if you win the game you offer a rematch; if you lose the game, you ask for a rematch. So in my opinion, anyone of both is able to push the button of rematch.

I might offer for live, although in turn-based, it is a completely different story. I rarely offer a rematch, although maybe that's because all of my games are team matches or tournament games.
There is only one circumstance when I feel somewhat obligated to offer a rematch. (I say "somewhat" because if it's late and I only want to play one game before bedtime, I still won't do it.)
If I am playing a player with my rating or higher, who loses a game because of a boneheaded blunder, like a hanging queen, I will offer a rematch. I won't do it to a lower rated player, because if they are rated lower than me, that might be the sort of thing they do all the time, and I don't want to be seen as piling on. On the other hand, if they offer a rematch in those cases, I will almost always accept.

I think what most people have already said is substantial enough... it doesn't matter who offers the rematch just as long either player doesn't get hung up on a decline. It would be nice for old customs to return though...such as "the winner offers a rematch"

Is there a protocol for who should offer rematch, loser or winner?
No. Why do these inane topics keep resurfacing? Can't we talk about something interesting like space aliens?
@ electric pawn: these 'inane' topics keep coming up because every now and again we want to turn our minds from the strict rules of chess to the fuzzy logic and irrationalities of social behavior.
And what in the forum title (Who should offer rematch?) made you think you would find aliens here? ;-)
@ trysts (or anyone): what does BS mean?

My oppenents always ask for a rematch when they lose playing black side, and I always accept it. But I don't understand when they decline when I ask for a rematch in the same situation...
the community is loosing some manners... I'm not even talking about the badmouthing.

If I lost and want revenge, I offer a rematch.
If I won and want to play another game, I dont offer a rematch, because I feel like I might intimidate my opponent into another game.
I believe it should be the loser to offer the rematch. It is bad ettiquette in general to beat someone at any game and then imediately offer another game, the losing player could interpret it as "rubbing one's face in it" or a want to humiliate the other player. I hear ye cry "it's just a game!"... games are a simulation of life and we should respect our fellow man/woman and give them every oportunity to recover from their loss before returning to fight another day.
A winner should offer rematch, it's like a good manner but not the rule. But in on-line chess people forgetting the traditions of the chess :)
It's got nothing to do with manners.

you can write a letter to fide and ask for a rematch
otherwise you probably will have to participate in the candidates cycle again
you should taken the opportunity when you had it and beaten your opponent
If I feel like I easily outplayed the person, I wont offer a rematch because I'm usually looking for a tough game / I'm trying to get beat :) But I wait 5-10 seconds and if they ask for one I'll accept. I only ask for one when I lose and I feel like I could have done better or they played in an interesting way. So in my opinion if you want another game it's best just to ask even if you're the one who lost and if they deny it no hard feelings, there are plenty of players out there.
I guess I haven't been around long enough to be aware of the etiquette that only winners offer rematches.