If the two sites were using the same ASP.NET membership store, sharing the same domain name (like www.mysite.com and community.mysite.com) and both using Forms auth then it would work by making sure the sites use the same authentication cookie name and have the same machine keys.
The other option I mentioned about setting up the www.mysite.com site as an OAuth client for the Telligent site could also work and it would cover your requirement about not having to manually login on www.mysite.com once logged into the Telligent site. However the most likely scenario would be that users get a typical OAuth approval page: Do you want to allow MySite.com to act on your behalf" sort of thing. You could get around that part of it by using the Client Credentials authorization grant type but you would still somehow have to determine who the user is to be able to get an auth token through the Telligent site. We do have new OAuth client functionality slated for the next release that doesn't require that acknowledgement page if the other site is a "Trusted" client, which in your case of it being two of your own sites it could be. But you would need to write some custom code to handle getting the access token from the Telligent site and then auto-logging them in on the www.mysite.com site.
Otherwise I think you're going to have to either look into some sort of single sign on system.
In any case you'll have to write some custom code to perform this functionality. There is nothing OOTB to handle this. Our Services team could help.