First let's consider a saddle. on any flying animal you will want a saddle that actually attaches you the the animal, you are not just sitting on it you should be strapped to it, you don't want to come loose during maneuvers, which a flying animal will do in all three dimensions.Your riders should be so secured that even if they are killed they will not fall off. This makes things like lances or catchpoles pointless, you'll break your arms or just end up bringing both mounts together an a fouled mess before you dislodge them.
If the animal is much much larger than a person (something fantastical) you can get away with the person being upright but for anything else you have to worry about drag. you also can't put your legs down their sides becasue the wings are there, giving you three choices.
the rider can be lying flat against the animal, belly down, to reduce drag. that reduces what you can use. this means you have very few good angles of attack, the wings get in the way in the downward direction, the head is in the way forward. Up is you best bet but also the most difficult for a human to attack from. Your riders may actually have a saddle designed to let them turn sideways (effectively lying on one arm) so they can reach out.
another option is to be saddled in a kneeling position and stay in a prone prayer like position or lying flat most of the time, only sitting up to strike. this would let you use a wider range of weapons (even bows) as well as give you a wider angle of attack, including letting you lean back to strike upwards.
the last option is to have the rider legs forward and lying back belly up, lifting the head to see, like a bobsledder. This is the most complex position, and requires the most confidence in your mount steering itself, but also the one that gives you the most options for attacking others, combining this with option one in a tandem, two rider, setup would also work well, one focused on steering the other on attacking. this does double your weight however.
There is not much point attaching another rider when the mount makes a much better target, especially the mounts wings. Wings are fragile, essential, and large targets. Any slashing blade on a stick(like a naginata) will work well for this, even swords although less effective. Slashing at the enemy wings will be most effective but is also the most dependent on winning a competition of maneuvering almost like a dogfight. Alternatively a hooked or barbed disposable weapon would also work well, something that can left behind and will stay in the wing, fouling it while doing even more damage. One large hook or spear in a wing will take an flying animal out of the fight. even if it doesn't kill it.
If you use option 2 or 3 you can use bows, composite shortbows like the mongols or crossbows would be best, barbed arrows to stick in the wings of your opponent would be best, that way you don't have to hit the body to do damage. A trained group of flying archers would be devastating against ground troops just look at the variety of maneuvers created by horse archers and how devastating they were.
You can also hang a line of barbs of hooks off the back of the animal almost like a troll line dragging behind it. it will need some kind of break away so you don't end up fouling your own mount if you catch something, but a line of barbed hooks will injure rider, injure the mount, and likely bind the wings, which would be fatal. They will be costly and heavy limiting how many you can carry but they protect your underside and have a much large strike range. I could even see these used with kites both on the ground and anti-air and from mounts to increase the range.
weapons you don't want to use.
Nets need to be swung, and if you miss your in danger of tangling you own mount. if drug behind you get a less effective version of the hook line for more weight. carrying between mounts might be possible if you have it furled, but that's a lot of weight and drag so you need a big mount.
Lances or halberts, besides the weight, unseating another rider is unlikely and requires either your mount to fly upside down or incredibly precise flying to hit. both are designed to take advantage of horse saddles not ones that actually secure you to the mount. Spear like lances can work although they will be difficult to use and again you will need to let go. Any of these will need to be much longer than normal to reach the other rider.
Maces or flails, you run the risk of hurting your own mount as much as the enemy, you don't have the wide open side for striking as you do on a horse, the wings get in the way. One missed swing could bring you crashing to the ground when you hit your own mounts wings. Anything you swing that has a large recovery is right out for the same reason.