Invite feature works as designed; I just tested it on try.discourse.org – everything hinges on being a trust level 2 user, as described in all the Discourse documentation.
I signed up with new account (all accounts default to trust level 1 there). I can’t send invites as a TL1 user. The share button on the topic does not present the option, and there is no invites tab on my user profile. Like so:
However, once I granted the new account trust level 2, notice that the share button on the topic now offers the invite option:
Invites can be sent via the invite dialog
Furthermore, as a trust level 2 user, I can track the status of all my invites via the invite tab on my user profile:
Perhaps you have some other site setting enabled that prevents invites from working? Is your site set to private? (For example, SSO / DiscourseConnect used to prevent invites from working, but I believe this is no longer the case.)