Have the same rendered output, whether the video is a local or remote ressource I need to let contributor either upload video or link to some remote streams (youtube). 

For this, I use media (and media_youtube) module. 

I works well for remote ressources, however when I try to upload a video instead, the video appears as a link, on the front (though I tried to fumble with the manage display settings for this peculiar field). 

I tried to implement jwplayer for the local videos, but it did not work this time. I also had a look at mediafront module, but as far as I know, it does not handle "local" videos (I may be wrong, but it seems overkill). 

Have you been facing such a problem before, that is render a consistent display (a player with the same dimensions), regardless of the source (local or remote) ? 
Can you point me to some resources ? 

Thank you.
Haven't tried it yet, but I believe that is what is good about the mediafront module.  It wraps the youtube or local videos in it's own themeable player.

You could also try http://drupal.org/project/mediaelement module, though I'm not sure how well that display's youtube videos.