Thank you for using Apple Support Communities. It's my understanding that you're able to successfully send messages from your iPad, but you're unable to receive messages. I can surely see how this could be a bit confusing. I'm happy to help.
First, I recommend checking your internet connection. If the issue persists, restart your iPad. Restarting can resolve many unexpected behaviors. Restart your iPhone, iPad, or iPod touch
If the behavior continues, let us know if this occurs with multiple networks and all contacts. Please include any additional troubleshooting steps you may have tried.