Draw offer

Whose clock is ticking when opponent is thinking whether to accept the offer or not?

It depends whose move it is. Which ever side is to move will have their clock running whether or not there is a draw offered.

Then it seems that the fair thing for me to do when offering a draw, is to do so when it is my turn...

FIDE rule 9.1

A player wishing to offer a draw shall do so after having made a move on the chessboard and before stopping his clock and starting the opponent`s clock. An offer at any other time during play is still valid, but Article 12.6 must be considered. No conditions can be attached to the offer. In both cases the offer cannot be withdrawn and remains valid until the opponent accepts it, rejects it orally, rejects it by touching a piece with the intention of moving or capturing it, or the game is concluded in some other way.

Fide rule 12.6

It is forbidden to distract or annoy the opponent in any manner whatsoever. This includes unreasonable claims or unreasonable offers of a draw.

Offering a draw on your turn allows an opponent to wait until you run out of time. Another opponent will want to see if you blunder or make an amazing move to determine whether the draw offer should be accepted.

Thanks TadDude.

Considering "Offering a draw on your turn allows an opponent to wait until you run out of time", is there a time limit within which a draw is to be accepted, else considered rejected? Or do we trust good manners and a sense of fairplay of our opponent?

I stated it badly. Offer the draw on your turn but as indicated in rule 9.1 you make your move, now offer the draw while your clock is still running, then start your opponent's clock so now it is his turn. Your opponent has until just before his flag falls to agree.