Is it normal to always tear up more in one eye?

Ok, the joy and pain thing is untrue obviously. The reason your right eye probably tears up more is likely due to a clogged puncta which is where the tears drain out of your eye and into your throat. If it is clogged in your right eye, then you have more tears that aren't getting drained and build up and drip out of your eye. Either, that or your right eye is producing more tears probably caused by irritation from contacts, something stuck in the eye, etc. There can be many reasons for this. This is pretty common and is easy to treat.

If it becomes a problem you should see an eye doctor.

The most common reason our eyes tear is due to dry eyes. Our tear film consists of 3layers each different from the other and if one layer lacks in production then the other 2 will try to compensate by over producing. Use artificial tears in your eyes everyday. 3 to 4 times a day and if that doesnt solve the tearing add a tears ointment a bedtime because thats when our eyes dry out the most because we are not blinking. Dont worry yourself about over dosing your eyes with tears unless you worry about overdosing your skin with lotion because its the same concept. I wouldnt consider a blocked punctum as the problem because that is not something that just happens over night. Other than infants that are sometimes born with punctal occlusions the only adults ive ever heard of to have a occluded punctum are thoses done intentionally by a ophthalmologist after a patient has gone through the necessary means to have them blocked to aid in treatment for severe dry eye syndrome.