By the FIDE rules, one can accept or decline a draw offer at any time - but making a move counts as declining it. As long as you haven't accepted or declined the offer it is outstanding, but you can't play another move without making your decision at the same time.
On this site, draw offers are handled exactly the same way. The "accept/decline" prompt appears for your opponent if you clicked "Offer draw" regardless of whether it's your move or his, and it simply disappears if your opponent makes a move without choosing either option.
I am relatively new to this site. If I offer a draw, does my opponent need to accept or reject the offer on his next turn or is the offer merely outstanding. Thank you.