In a true divided system it cannot go from side to side. However, In my case when they c-sectioned my son out of the right side... They cut horizontal which made a communication between the two horns... Which in return led to 8 miscarriages cause when the pregnant side got to big the baby wanted to pass through the opening which caused water to break around 23 weeks... Dont know if you have had a c-section yet but would def aim toward a vertical cut... As far as knowing which side... Did you ask your Dr. if there is a way for him to detect on an ultrasound during ovulation times so then you can try and relate symptoms with which side you are ovulating out of?
On an ultrasound some day before ovulation your doctor would be able to see the "leading follicle", that is the follicle that will release the egg at ovulation, and on which ovary it is on. The egg is probably more likely to go into the tuba on the same side, however it is entirely possibe that it'd float to the other side. So, no there is no way to be sure the egg has passed into the "right" side.