After 145 games, I have never offered a draw in Online Chess. Seven of my draws are "by agreement," and my opponent offered the draw each time. The other five are draws by repetition.
Several days ago, I briefly considered offering a draw in one game in which I could not figure out a plan. I soon decided not only that I should not offer the draw, but also that I would never offer a draw in an Online Chess game.
The reasons are numerous and obvious. First, I want to win as many games of Chess as possible--after all, the object of the game is to win--so why should I offer a draw--something I don't want?
Second, should I offer a draw if I think I might lose a game? Well, wouldn't that be cowardly? If I'm going to lose the game, then I have no business offering a draw; I deserve to lose, unless I can outplay my opponent from that point and save the game, turning it into either a draw or a win.
Offering a draw would be a signal that I want to end the game immediately. I would essentially be saying, "I don't want to play [with you] anymore." If the game is supposed to be drawn, my task is to prove that by executing the necessary moves--and I will make my opponent do the same, if necessary.
I like Jeremy Silman's counsel on draw offers. He recommended rejecting almost all draw offers (except when the game has reached doldrums) and never offering a draw.
If I'm playing someone lower-rated, I should be expected to win, so offering a draw would be a fearful concession.
If I'm playing someone rated comparably, I stand to learn much from the game. Offering a draw would be a refusal to learn more about my skills and the Game in general.
If I'm playing somoene higher-rated, I again stand to learn a great deal from the game. Additionally, the draw offer could be construed as an insult to my opponent.
These guidelines help me approach draw offers from my various opponents as well. Though I am never offended by draw offers from my lower-rated opponents, I do understand that an element of fear may be governing the decision.
Clearly, I am more lenient regarding accepting draw offers than making the offers myself:
1) The first draw offer I accepted (OmegaSupreme, 1869) because I had a Rook and two Pawns against two Bishops and a Pawn, and I couldn't separate the Bishops in order to try to skewer them. I could not make progress.
2) The second game (tapirus, 2046) was a tournament game. My opponent had a material advantage and a much better position. I was incredulous that the draw was offered to me. I accepted it because doing so ensured my advancement to the next round. Had I lost the game, I surely would have been eliminated.
3) I had a severe material deficit in this game (pawnkiller2, 1815), but I elicited a draw offer from my opponent because I had a Pawn on the seventh rank with a Rook behind it, tying my opponent's Queen to blockade duty. From the final position, White had an easy win (+8.60). My opponent perhaps could not figure out how to defuse the promote threat and attack.
4) In another game in which I had a material deficit and a bad position, my opponent (clivebeesley, 2096) offered a draw. I do not know why, although laziness may have been a factor, as White's victory appears inevitable.
5) In this tournament game (kswalters, 1803), my opponent offered a draw twice. I rejected the first but accepted the second. Though I was initially determined to tangle with my opponent for the long haul, wear him down, and eventually win, I realized that in the final position, my opponent may have had a way to force an ugly position on me. If I had continued 29. ... Qc7, intending 30. ... Bd8, he could have played:
This game is, to this day, my only non-win in the first round of any Chess.com tournament.
6) This game (Mainline_Novelty, 1647) had reached a King-and-Pawn endgame where Black's weaknesses are either very difficult to reach or easily defended.
7) Finally, this game (meh_go999, 1888) was another instance of having been let off the hook. The game could have continued:
Though I don't regret accepting any of these draws, my philosophy toward draws has evolved. If I had the opportunity again, I would have rejected the draw offer in the first game in this list and continued playing. Of course, if I'm offered a draw when I'm obviously much worse (which fits the descriptions of the second, third, fourth, fifth, and seventh games), I will accept it. I'm still undecided about the sixth game. Though I was not losing, the game would most likely have been drawn anyway, and the likelihood of blundering meaningfully in a King-and-Pawn endgame is comparatively very high.
So I will in the future regard draws and draw offers with much more stringency. I may continue games for much longer than my opponents would prefer. I will not do this to offend them; I just want to win. After all, Chess is a competitive game.