Edited on January 25, 2015. Added video instructions.
As of right now, every time you boot up Destiny your Fireteam is automatically set to private. In order to open it up to the public (which everyone should do if not playing with friends) you have to complete a few steps. Every single time you decide to play. That’s right, every time you quit out of Destiny your Fireteam settings default back to private.
These instructions are written assuming you’re not using a Kinect (I didn’t bring mine with me overseas). I would assume you’d be able to cut out a few steps using Kinect voice commands, but I’m sure you can figure that out.
So the first thing you want to do is hit the Home button.
Next you’ll want to select the panel shown above.
Then select My Profile.
Select Destiny (off to the right assuming you’re currently in game).
This is what you’ll see before you change the setting. Select ‘Requires invites’ to change it to ‘Set to joinable’ and you’re good to go.
avideogamelife 
I don’t have that “join in” section, only activity feed… what do I do?
Is Destiny open (and I think you have to be logged into the game)? It should show up to the right of the activity feed if you’re in the game. If it’s not there then I’m not sure what’s going on, sorry.
This is just the worst system I have ever seen… Wouldn’t it have been easier to keep the exact same settings system for 360? You go through one menu (maybe two depending how you consider it) and one little button. This is just too much…
This is just the worst thing I have ever seen… Why wouldn’t it be the same as it is on the Xbox 360 version? It’s one menu (two depending on how you consider it) and two button presses to get into the menu and decide if you want to be open/closed/invite only. This is just stupid…
I can not send invites for people to join my fireteam every other button works except the invite to fireteam one can anyone help me
I would suggest checking your NAT, but I’ve never encountered that issue. Maybe ask on the Destiny forums.