Typically, that info is mentioned on the Realist report. That is the public record database that Sandicor subscribes to.
Honestly the biggest complaint I have about the private consumer mls services in this area (sdlookup etc) is that they don’t have this sort of access.
Also, there is a field for this on the mls but it is sometimes left blank by the agent.
That is because agents get sued sometimes for stating the school district incorrectly (whether by mistake or on purpose). Not such an issue in Hillcrest or North Park or Downtown (where I do business) but a really big deal in suburban communities. If you thought you were in the Encinitas district and turned out to be in Vista,
it could really affect your decisions for buying.
The single most reliable source I have seen for this is the websites for the school districts themselves.