Nope, not possible: it’s not really in scope for the main website which is really a portal for editors (although still useful for map browsing).

You can upload GPS traces in private mode which will be added to the overall public pointcloud which is used to check alignment of imagery & add totally new roads & tracks which post-date available imagery. However, I don’t think this is what you want, although if you are walking in an area where OSM lacks detail it may be generally useful.

I think I’ve used GPSsies in the past (see OSM Wiki page https://wiki.openstreetmap.org/wiki/Category:Tracelog_hosting)) . This has close links with OSM. I’m not entirely sure if you can keep a track completely private though. For smaller scale specific projects (e.g., visualising a walking tour) GPX layers can be imported to umap.openstreetmap.fr and the map kept private.