The question is whether you're training in Muay Thai to compete in Muay Thai matches, or are you training in Muay Thai for MMA (or other)?
If you're training in Muay Thai for the sake of Muay Thai, there are specific ways in which you can legally throw your opponent. You cannot throw your opponent over the back of your leg, hip, or shoulder. You cannot use your leg to "hook" your opponents leg to take them off balance. You cannot use the inside of your foot to trip them either (I believe in Judo you refer to this as a reap?).
To properly throw in Muay Thai, its a twisting motion. You essentially twist your opponent like a steering wheel. While performing this motion, you can twist them over the INSIDE of your leg, but you MUST step back and clear the leg out of the way before they fall to give them the opportunity to regain their balance. Otherwise this is ruled as an illegal trip.
For instance, you and you opponent are clinched. You step forwards with you right foot outside of his left foot, blocking your opponents left side. You twist him over your right leg so that he begins to fall. Before he finishes falling, you must step the right leg back out of the way or it is considered an illegal trip.
If you want to kick their legs out from under them, you have to kick their leg/s out with a proper roundhouse kick impacting with the shinbone or instep. So hypothetically if you were to trap your opponents kick, you would want to kick through his support leg to take him off his feet. If you try to hook his leg or sweep with the inside of your foot, it is again considered an illegall trip.
If you're training for MMA, then you can pretty much throw them however you wish providing you are not "bulldogging" (driving opponent head/neck first into the mat) them in the process.
