Which one would a native speaker use?
compete against? or compete with?
e.g.) We didn’t want to compete [color=red]against/with them because they were professional at playing football.
Is this also correct?
We competed with each other.
In that context I’d use ‘compete against’ - you are talking about opposing teams, possibly taking part in a tournament.
I’d use compete with where more of a spirit of co-operation is needed in order to achieve the goals.
To me if you compete with someone/ are in competition with someone:
you have a shared recognitions that the other is a valuable resource, essential to our progress, achievement and long term health.
Although there is competition there is also co-operation.
When we compete against / are in competition against others, there is more of a sense of an enemy to be conquered.
There is no requirement to cooperate for the common good.