Well depends:

- if you are good in programming or if you have enough time,
mathematical mind and will to learn it. Then you should learn
ActionScript 3 (I think its useless to start learning AS2 at this time)
and expand designer (illustration, designing, typography...) skills.
Then you can be a flash developer and designer.

- if you are not good in programming and don't even have time for
learning it then continue learning about design, flash animations,
everything in flash that doesn't require a lot of code. Then you can be a
flash designer.

To be competent and get more clients most important thing is to do good
work. If you can't be good at both developing and designing, find where
are you better at and go with it. Specialize!

Good Luck!

Well, not sure it's "required", but I think knowing some will certainly be an asset.

I've been working with flash a lot as a "designer" (doing animations, banners with just 1 getUrl link in it, etc). I always solved problems in a way that required almost no programming for me. This often took me a lot of time to do simple things. Now I've decided to learn at least the basics of ActionScript 3 and it's been a life saver.Now I can do much more with flash (wich means extra work, extra income!) and I solve problems 10 times quicker because I understand the code much better now.

My advice: Start with the basics, like making banners or a simple website and do some tutorials on ActionScript 3 basics alongside.

You dont need to know every thing about AS3, the important part is to know how to solve problems, and have a general idea about what can flash do or not, after that, solve specific problems will be easier