This is really a personal preference. I log after every shot and then forget about the shot and the hole score. Others enter all shots for a hole between holes. Finally, many enter after the round. This requires a few notes on a scorecard (we don’t have a special scorecard). You can find an example of these ‘notes’ here: https://twitter.com/Fit_For_Golf (not sure exactly where in his feed it is, since he tweets a lot).
Downside of during the round: Some people think during the round interferes. (Me: I log shots instead of using a scorecard.)
Upside of during the round: No note needed, report available immediately after the round, no time wasted entering data post-round.