Register directly on subsites?

That was never a feature of basic WordPress. You might have had a plugin for it, but bog standard is that everyone signs up and is a member of the main site.

You would need to use the plugin if you wanted that feature.

https://www.ads-software.com/extend/plugins/private-wordpress-access-control-manager/

It seems the plugin doesn’t have that functionality . Or I am missing some thing.

I have also edited private.php in the plugin files.I have made a post here for plugin developer .

There is a plugin called User side bar widget .but that too works partially.
Could you please suggest me some thing to make this functionality work?

Thanks !

Copy wp-signup.php into a page template and use it on the theme in the sub blogs, removing the redirect to the main site at the top of the file.