> otherwise both of the names will render? 

Depends on the renderer, but in general, yes. A name will render for every object given a name. Trail names in bridges make it look like the bridge has a name of its own. Even more confusing, a search for a name will return every object given that name. It can be hard to pick out a trail's overall relation amongst the noise. 

> Just leave it blank or label the path with a local trail name? 

When I do not have the time to create a relation and document it on our Michigan trail project's wiki page [1], or the trail is short, I will add the name just to the trail's largest segment. Naming one segment is enough to make the trail render, be searchable, and show amongst routing instructions. And, I only name a thing after I have found its name on a sign or official document [2]. 


[1] https://wiki.openstreetmap.org/wiki/Michigan/Trails 
[2] https://wiki.openstreetmap.org/wiki/Good_practice#Don't_use_name_tag_to_describe_things 