The trustee needs to sign whenever there will be changes made to the trust. If your question is; does the second trustee have to sign in person at the bank while opening a bank account, it will depend on the bank. So you need to ask the bank you will open the account at if the second trustee needs to sign in person. However, our experience is that most banks will allow for sole signatory control.
As to signing other types of trust documents, no, signing them in person is not required. It can be done via paper post or electronically — whatever is acceptable to the trust officers involved.