I left mine in place & stripped the paint the same time I removed if from the surrounding wood. Same thing with the sash pulleys. Re-paint the wood first & then use an artist's brush to prime (if necessary) & paint the hinge.

If your hinges are screwed in as tightly as mine, it would have been far more damaging to try to remove them.

I wouldn't replace them but if you want to clean them so they are less caked with paint you could try putting them in a old pan that you will never again use for cooking and boil them. That should soften the paint up enough that you can remove it. You may have to go thru the process a couple of times depending how thickly they are caked.
Once clean you should be able to see if they need a rust protector and I would spray paint them what ever color you want to go with your shutters or frame work.

I'm not sure if you want to take them off but if you do here's a great easy way to get the paint off (I'm doing about 20 interior doors). Get a slow cooker you won't use again for food. Set on low and fill with water and a few drops of dish soap. Soak the hardware over night. Pull out the hardware with tongs the next morning (very hot) and the paint just slips off it. I use a brush to help get any more paint out of in the crevices. Some of my hinges have been painted for 185 years.

Ours look pretty much the same.. but it will be a while before we can get to them. I plan on using a heat gun to gently remove the paint from the screws, so that I can unscrew the hardware, than I will do the same crock-pot method (I use laundry soap).. works wonderfully!