No, your trustee only has whatever information you give him or her. If you are manager of the trust, that role includes the ability to bank, be the signatory on bank accounts, add assets into the trust, take assets out, and you can organize it so that the trustee is informed of, and signs on, only those transactions on which you wish to include him. If that is “none”, then it will be “none”. It’s up to you.
This is much stronger than an NDA. A trustee or trust officer can only take what he knows exists. If he doesn’t know it exists, there is no way he can even have the thought of taking it.
For example, we have a friend as co-trustee with us on a couple of trusts, but he hasn’t the foggiest idea what these trusts have. Nor does he have access to those assets, because he lacks the information about them. And we would trust him with our lives . . . but after all, he wouldn’t even WANT to know what the trusts have . . . or otherwise, if he did, and if anything were to happen to those assets, he could be erroneously suspected. So he doesn’t even KNOW about them.
Normally a trustee is signatory on trust bank accounts, but as manager you can organize it so that the trust only authorizes you, not the trustee(s), to be signatory.