When I send a Private Message I get a message that says "private message sent", then the message is in my out box, but I don't see it in the sent box. So I don't know if people are really getting the Private Messages I sent. These are replies to peoples questions and I don't want them to think I'm ignoring them.

Is there something I should be doing to make sure they get it, or how do I know it's been received?

If something is still in the outbox it means it has not been picked up by its intended recipient yet. Once it has been picked up it will show up in your sent box.