The usual procedure with territory scoring is to first agree with your opponent as to which groups are dead and which are alive. Then, remove the opponent's dead stones from your territory, and not your own from theirs. This avoids the possibility of confusion between removed stones and your own stones.
You should first remove all dead stones from the board, before you start filling in territory with prisoners. Mixing this up may lead to confusion about the status of stones when territories get rearranged.
After all dead stones have been removed, fill in and rearrange the territories of your opponent, while they do the same for your territories.
See Japanese counting for details.
With Chinese style rules, there can be no confusion over prisoners, as prisoners and dead stones do not count towards the score. All dead stones are removed and returned to their owner's bowl.
Make sure you correctly mark all stones that are dead as such before letting the computer calculate the score.
Mark you own dead stones first, and, not at lightening speed. Give your opponent a chance to watch what your are doing if they want. If your opponent has not even started marking their dead stones when you have finished marking yours, insert a slight pause to give them a chance to mark their own dead stones before you start marking their dead stones for them. Maybe they would prefer that you mark their dead stones too because they consider you to be the stronger player.
If they mark some (or all) your dead stones, probably best not to complain. Just be careful that their judgment is correct before you let the computer calculate the score.
See also: /discussion
 For example: Suppose you have a few of your own stones lying on the table on your side of the board (perhaps as a remainder of Canadian byoyomi stones for the last period). Then if you remove stones of your own color from the opponent's territory, and accidentally drop a few on your side of the board, you might get into a conflict over how many stones you dropped, and how many were already on the table. The only opposing stones that should ever be on your side of the table, however, should be prisoners from during the game. So if you drop a few dead opposing stones on your side, there is no problem, as they are the same as prisoners anyway.