GPS track public or not ?

All trackpoints from GPS traces that are uploaded to OpenStreetMap will be available to view in an editor, whether they are marked as Identifiable, Public, Trackable or Private. But only tracks marked as Identifiable or Public will show up on the traces page on openstreetmap.org. See Visibility of GPS traces.

If a trace is set is Identifiable, then the editor can link to it in the traces list. In JOSM, you can do this by downloading all GPS traces in an area, then right click on Downloaded GPX Data in the layers list, and click on Info. Any identifiable tracks will be listed with their URL, so you can then go to that page and download the original GPX.

For traces set as Public this is not possible. But you could manually look through the traces list and see if you can find it. Also try checking for any relevant tags.

For otherwise unidentifiable traces, you can still download them all into JOSM, then save the layer as a GPX file. But this will have all of the traces joined together, so probably not very useful.

You should explain what you are trying to do. If you can see a trace, it's public.