If you are only seeing received messages in Droid Transfer, this usually means that you are using Advanced Messaging (or RCS Format) messages.
You can confirm this by going to the Messages App on your phone; then long pressing a sent message to display the message properties. The message type may display "Advanced Message" or "RCS" message.
This is particularly possible if you are based in the US and you use AT&T or Sprint as your mobile service provider. Droid Transfer only currently supports standard SMS and MMS messages.
It is possible to turn of Advanced Messages in devices settings / options.