Same issue on my iphone this evening after updating to iOS 7.0.2. Sending an message appears to start, blue bar moves across on the menu bar above the message to indicate transmission, then pauses with about 10% to go, status remains as "sending". Happened on my iPhone 5 today, happened on my freinds 4S yesterday, exactly the same symptoms. 🙁
In my testing i attempted to restore both phones, then set up as new phones, signed in with AID, carrier settings were updated and icloud was used to sync contacts back, issue was resolved at first but then came back. Didnt notice issue until tonight when nobody was replying to my messages. Thought to myself, ok i may have pissed one person off, but i KNOW i havent pissed everyone off! Ended up having to tap on msg and use the 'send as text message' option.