Stuck - there are two features in DT Register that will force an event to be private. Both are stated in the tooltips and in many other posts in this forum...
1) Overlapping Event Check - when enabled all events in DT Reg will require login.
2) Prerequisites - when an event has any prerequisite event and/or category selected, that event will require login. Use CTRL-Click to de-select anything that you may have accidentally selected.
These are items that will cause the "private event message" from DT Register to show. Check the message showing on your site... is it the one from DT Reg or from Joomla that shows whenever something in Joomla is assigned as "registered" access??
If it's the Joomla message, then your issue is tied to joomla menu items as stated here by thepiston. DT Register will automatically find the first "com_dtregister" menu item in your database and use it for DT Register urls. If you happen to have one that is set to "registered" or "special", then that Itemid will force login to access the page. Sometimes users have created a "registered" menu item for the frontend user panel of DT Register, which makes perfect sense to do. That's how our joomlaeventregistration.com demo site is setup. If for some reason that menu item is being accessed by DT Register instead of the main event listing menu item, you could delete the existing user panel menu item, then re-create it. That should bump it down the list. Be sure to remove any old DT Register menu items from your menu trash too. Also, obviously, make sure you have a valid DT Register menu item for listing the events that is Public status. You don't have to have that visible on the site... just needs to exist so DT Register has a valid public Itemid to grab hold of when generating the urls.