Do I have to host my data on Woosmap?

Yes and no, it depends on the API and use case your are implementing.

For instance, if you are using our Store Locator to help users visualize the closest store on a map, you must indeed load your stores list onto the Woosmap servers.
On the contrary, if you only use Localities API to autocomplete postcodes or city names in a search box, no content is required from your side because the autocompletion is based on geographic data.