What should a player do when their opponent asks to see your notation sheet during a game? During an over the board tournament, an opponent frequently asked to see my scoresheet, stating that they had missed moves or otherwise messed up theirs. (They asked 5 or 6 times in a game with a time control of game in 25 minutes.) Trying to keep things friendly, I let them see my sheet. The first couple times I did not mind, but when it continued happening I found it annoying and distracting.
Thus the question: Am I obligated to let an opponent see my game record so they can make corrections to theirs?
Also, if I do show it to them, whose clock should be running as they make corrections?
Am I obligated to let an opponent see my game record so they can make corrections to theirs?

According to USCF rule 15D3, it's suggested that you do so, but it's not mandatory unless the tournament director tells you to:

The opponent is urged to comply with such a request, but this is not mandatory. If the opponent denies the request, the player may stop both clocks and see a director. A director who agrees that the request is appropriate shall instruct the opponent to lend the player the scoresheet. The opponent may not refuse as all scoresheets belong to the organizers.

The TD will probably allow it, unless there's some reason not to. Excessive requests may be denied and may even be penalized, according to rule 15D4.

Also, if I do show it to them, whose clock should be running as they make corrections?

According to USCF rule 15D2:

The clock of the player making such a request is running and shall continue to run until the scoresheet has been returned.

Under FIDE rules, the time control in question would be a Rapid game. According to FIDE rule A.3.2, which applies to rapid games:

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.