cross account access to s3 and transfer traffic

If account A has full access to account B’s S3. If upload and downloading objects happening in account A (on account B’s S3) , does account A need to pay charges ? Obviously Account B will pay for storage but what about account A charge for upload or download ?

The general question is if an external account access your AWS account and use your account’s services, is there any situation that the external account will be charged ?