i have created private repository with "devloper" access to other user.
i am using GitHub Desktop with GitHub account login in win 10,
i had clone repository wiith dev acc , at the time to push commit to repository i used git desktop which has login using "GitHub" user id not Bitbucket.
Even then able to succefully push changes.
At Bit Bucket Dashboard
Unmapped user This user cannot be matched to an Atlassian account.