Merchants don't need to share affiliate links with publishers through Skimlinks.
Skimlinks works by spotting natural merchant links and redirecting them when clicked. So, as a merchant, all you have to do is ensure you're set up on Skimlinks. Once that's confirmed, just inform any publishers you want to collaborate with that you're active on Skimlinks.
They'll either already be on Skimlinks and know how to create links to your program and others, or they won't be on Skimlinks yet. If it's the former, there's nothing more for you to do except let them know you're using Skimlinks. If it's the latter, you can suggest they join Skimlinks. Once they're in, they'll receive all the details on creating links and sending traffic to your merchant program through Skimlinks.