An account may send a token into a storage account when it has the right to deposit that token there: its own right for that token, else its own right for the whole storage account, else -- only when it has no right of its own -- what the storage account lets anyone do. Its owner and admins always may. The token must be one the storage account accepts.
An own right replaces what anyone may, so an account with a right to send on behalf of the storage account for one token, but not to deposit it, may not deposit that token from its own wallet.
A right this storage account grants earlier in the same staple counts; a right on another account does not. Some storage accounts let deposits in this way only: they grant the right, take the deposit and take the right back, all in one staple.
A token's own account sending its token is not checked: the network let the bridge's first 1,029 transfers into its storage account before that account accepted any token.