It depends on why your distributor is asking for the information.
Your distributor may ask for your publishing information when you onboard songs to them for recording distribution. As a Songtrust client, what (if anything) should you provide to them if they ask? It depends on why they’re asking.
If your distributor is offering sync pitching or are including your songs on a sync licensing platform, they’ll want your publisher information so they can make contact when a request comes in. Since Songtrust doesn’t manage sync licenses, you’ll want to include your own name (or the name of your publishing entity) and your own contact information, so you can manage any sync requests.
DSPs like Apple Music and Spotify don’t ask your distributor for publisher information, so if you’re distributing through them for digital streaming only, it’s unlikely that you’ll be asked. If they do ask, and you have a publishing entity name, you can enter that. If you don’t have a publishing entity, you can just enter Songtrust.
If you are going to be manufacturing physical product and distributing it through them, they might ask for your publisher so that they can request mechanical licenses (though in many cases, they will not do this if you are a self-released artist). In this case, you can supply your own information or, if you prefer, you can enter Songtrust.
As you can see, there is no hard and fast rule here unless your distributor is going to be offering your music for sync licenses, because publishing information is generally not required for distribution.