What's not clear here is the order of user actions, and what events are triggering your macros. Regardless, I think you will find that On Error Resume Next will not help, nor will turning off warnings. What might work is to cancel the second update and/or Me.Undo the record edit as I think you'll find it's a message that cannot be turned off. This makes sense since (I think) 99.99% of such occurrences would be important if not downright vital to be aware of. Unless someone else knows how to turn off this message, I think you'll find the answer lies in preventing it altogether. Since appropriate record locking level settings don't seem to be an option for you, either try the aforementioned cancel or undo, or you might want to try using a temp table and attempt to move the record to the main table. Cancel the operation if the key field is already there and delete the temp record. This assumes the db is split and each user has their own FE. If not, you need to relate temp records to each individual user so that you're not wiping out someone else's active record too.
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.