Stellar uses signatures for authorisation. Transactions always need authorisation from at least one public key in order to be considered valid. Generally, transactions only need authorisation from the public key of the source account.
Mesh has implemented this via Signatories on the account screen. Each Mesh user has a private key associated with their user profile, and adding a user as a signatory will allow that public key to sign transactions for the specific account.
Please note that since the public keys for users are currently stored by Mesh, the user being added as a signatory will also require a Mesh user role in order to perform and authorise any actions on the account.