When a record is in someone's basket, it appears as "locked" to other users - the icon next to it in the search/browse results shows no option to add or remove the record and if you mouse over it, it shows you the name of the user that has the record in their basket.
When you click on the record, it opens in read-only mode.
From there, you can click on the edit icon 
, and the record will open in your editor where you can edit it. Only do this if you are sure that the other person who has the record in their basket is not in the process of making changes, because only the last changes to the record will be saved. If the other person is in the middle of editing, and you go in and make changes and save your record, and then they save the record on their end, only their changes will be saved. (It is rare that two or more people will be working on the same record, that's why we set it up in such a way that you can still make changes even if the record is in another basket. When in doubt, you can always contact the other user and double-check).