It's not difficult. In fact, it's easier to stay on than in a regular saddle. The problem is getting OUT of the heads (the sticky-out things - the upper one is the "fixed head" and the lower one is the "leaping head".) You adjust your stirrup so that your left leg is snugged up under the leaping head - and when you get in a tight spot (or when you're jumping) you just squeeze your legs together and you are STUCK on there.
The whole trick to sidesaddle riding is staying STRAIGHT. The tendency is to allow the body to twist left, towards the legs, putting all your weight on your left seat bone. It will give you a backache! To begin with, you have to concentrate on keeping the hips aligned straight forward, and keeping your shoulders up and back, even slightly rotated backwards to the right.
The real problem is finding a saddle that fits you AND your horse properly. Horses used to be fatter and longer-backed - so a T'bred with sticky-up withers and narrow sides is going to be hard to fit. There are a lot of antique saddles offered for sale that are in bad repair and don't fit modern horses properly, as well as cheap Pakistani or Chinese made ones that are just junk.
So where can you get a good new sidesaddle? Everyone I talk to that has ridden sidesaddle agrees with you, you are stuck on there and it's more stable than it looks.