I will never send my opponent a message asking them to resign.
Having said that, I still don't see the point of playing on in a hopeless position. I agree that everyone has the right to do it, but it's rather annoying and unnessesary.
If two players rated under 1200 or maybe 1400 are playing eachother in and engame, and one is up a rook or queen, I can see why the other might want play on: There is a somewhat reasonable chance that one player will make a mistake, or perhaps they want to learn more about that position. This is perfectly understandable.
However, when two player rated above 2200 are playing eachother, it seems useless. Kids, lets face it, a 2200 rated player is not going to accidentally stalemate someone. They just won't. Even at my level, which is about 1800, there simply such a miniscule chance that someone will fall for a one move stalemating trap that it seems useless.
Obviously, there are a number of positions in which a material mean advantage means nothing. However, I'm talking about the average position where a player is a piece or two down with no spectacular drawing resource.
I'm not talking about positions where one side is a knight down, but most of the pieces are still on the board, both players have somewhat of an attack, and both still have chances. In such a position it is perfectly appropriet to play on.
No, I'm talking about this type of position:
Sure, black could play on for 25 moves; But Why? What does he have to gain? At any level over 1300, there is virtually no chance that he will outplay his opponent to the point where he can win or draw.
Lets assume both players are rated 1500.
Black is perfectly within his rights to play on, so he does. There are a few things that could happen to make him win or draw:
1. He wins on timeout, probably the most likely. Playing on in the hopes of this happening seems rather useless to me, since most players want to learn to play better chess, and a timeout win will only stimulate his rating. While this will feel good, his rating will fall back to where it belongs withing a few weeks later.
2. His opponent has a mouse slip, and loses his rook, resulting in a possible draw. While black will get that warm fuzzy feeling inside, it certainly won't make him play better chess.
3. He purposely drags out the game and only moves every 21 days, hoping his opponent leaves. Eventually, white, who only has six open slots for games, and doesn't want to play this game for another year, quits. Once again, black gets that warm squishy feeling inside, knowing he was a better player than his opponent.
Personally, I feel it would be silly to play on hopeing for one of those things to happen. Even so, it is not OK for white to ask black to resign.