@GreenandBlack OSM doesn't allow data from authoritative sources (IP issues I guess?) so things like schedule and route data would either need to be crowdsourced or added in a third party app. On Android, for example, there's Transportr, which provides transit navigation using OSM for maps.
To be clear, Transportr uses OSM for maps, but not navigation. Navigation is provided by Navitia, a free (gratis) but not free (libre) service which uses a select set of available public transit services. Navitia doesn't take public contributions, instead you post a request to add a transit service, and a link to the data. They'll take a look, and decide whether they can/will. I was lucky and had my city approved, but my friend's was not.
I actually had an idea for an app related to this problem. So most transit agencies provide the necessary files (they have to make them for google anyway, might as well make it public). The reason Navitia doesn't host everything submitted is 1) specific licensing, and 2) server costs. So what if we implemented the Navitia API *in* Transportr, and download the schedule files from the transit providers directly? Alas, it's not so simple...
Navitia API calls don't require downloading the entire schedule again, they just cache it and run the navigation algorithms on their copy. If everyone started downloading these (sometimes ~100s MB) files from transit agencies to their phones, they'd likely get annoyed and stop making them public. :/
I still think there's the possibility for something interesting in this space tho, both for privacy and community control