This answer is about the FIDE Laws of Chess effective from 1 January 2018.
There does not seem to be any explicit provision for a player updating their scoresheet using their opponent’s scoresheet, except where exactly one player has stopped recording moves due to low time (Article 8.5.2):
If only one player has not kept score under Article 8.4, he must, as soon as either flag has fallen, update his scoresheet completely before moving a piece on the chessboard. Provided it is that player’s move, he may use his opponent’s scoresheet, but must return it before making a move.
- The player updating their scoresheet must have the move, so their clock should be running (assuming their opponent remembered to press it!).
- This would not apply to a 25 minute game (see below).
I would expect that requests to see the opponent’s scoresheet, or at least excessive requests, would be considered distracting. This is supported by the “rapid chess” rules, which would apply to a 25 minute game. See Article A.3.2:
The player may at any time, when it is his move, ask the arbiter or his assistant to show him the scoresheet. This may be requested a maximum of five times in a game. More requests shall be considered as a distraction of the opponent.
If asking the arbiter or their assistant six times in a game is distracting, then asking the opponent six times is definitely distracting.
You are definitely not obligated to let your opponent see your scoresheet if you don’t have one. In “rapid chess”, players do not have to maintain a scoresheet (Article A.2):
Players do not need to record the moves, but do not lose their rights to claims normally based on a scoresheet.