Are there any downsides to having wheels balanced with the weights on the inside of the wheel instead of the outside?

nope no problem unless you just dont like them on the inside...

no issues

I want to put them there because I don't like the way they look on the outside of my nice wheels. Just wanted to make sure that it wouldn't cause any problems.

Thanks for sharing your expertise.

make sure they are using stick on weights towards the outside of the wheels, but not on the lip if the rim, Otherwise, its known as "static" balancing, and they will not be very well balanced.

Ask for Dynamic Balancing with stick on weights