bitcoin core – What if two people want to spend same input at the same time

Lets say me and my friend both have access to same private key which can sign some new transaction. What would happen if me and my friend broadcast transaction with same input at the same time to a different nodes ?

Also what would happend if i broadcast it 1 minute after him and pay much larger fee and his transaction is still unconfirmed? Would miner take my transaction than because it includes higher fee or would the stick with his transction because he broadcasted it first.