The hosted ones still have the advantage of being able to search the internet for live info rather than being limited to a knowledge cut off date.
That's not how it works. Whether local or hosted, every modern model has a cutoff date for its training data, and can be leveraged by agents / harnesses / tools to fetch context from the internet or wherever.
Local ones that support tool use can do the same
You can do that locally too!
I’m not sure why a model needs to be hosted in order to make network calls?