Can a trustee of 2 separate trusts complete a telephone or online transfer between the two separate trusts? The irrevocable trusts are set up for both her parents with her as trustee on both trusts.

From a legal standpoint yes. I assume the trustee is authorized to conduct transactions for each trust. The trustee does have a fiduciary responsibility. In conducting the online or telephone transfer is a difficult question. I would assume, these accounts have different CIF numbers. This may be more of an account coding issue.

They each have their own tax federal ID number. I guess we are wondering for the allowable transfers since there is not a common ownership of the two accounts only the trustee is common between the 2 accounts is it allowable to make such a transfer

IMHO whether it's allowable and whether it's a good idea are two different things. If there has been a fiduciary breach, the bank has assisted and could be alleged to have been a party to the transaction(didn't say guilty of - said alleged to have been). You can better protect the bank by requiring the Trustee to write a check to make the transfer. This serves to remove the bank from the picture and creates a better audit trail.

I think you need to review the trust agreements. With this being said, some believe a bank should never keep a copy of the full trust agreement. Generally, a trustee has the ability to perform any type of transaction in connection with a trust. I am surprised your system will allow transactions from different ownership accounts to have transfers between them.