I stumbled on how to share in the FaceBook app. There were some complaints about this feature being missing. Since I can't remember what thread they were in or whether it got answered, I thought I'd just post my observations here.
First, I don't claim this is as convenient or versatile as it should be, so don't blame the messenger.
When you are looking at something you want to share, go ahead and follow the link, whether it's youtube, a website, or whatever. At the top right of the embedded web browser is the usual action/share icon (a box with an arrow coming out of it). Tap this and you'll see a share option. Add your comments and tap the Post button. There are some privacy controls at the bottom for fine control of who sees it.
This only works for websites. You can't share a comment from one of your friends, or their pictures, etc. You'll get a whole different set of options if you try, like Send a Message, Poke, and the dangerously red Unfriend.