The cleaning procedure is the same for either solvent. Wipe out the container and put in the magnets to be cleaned. Pour in enough solvent to cover them, and stir everything well. Let it soak for a total of maybe an hour, stirring occasionally. Pour off the dirty solution and check the magnets. If they still feel like they aren't clean, remove the magnets and wipe the container. It wouldn't hurt to roll the magnets around in an old rag to remove some of the residue. Then put the magnets back and repeat.

The cleaning procedure is the same for either solvent. Wipe out the container and put in the magnets to be cleaned. Pour in enough solvent to cover them, and stir everything well. Let it soak for a total of maybe an hour, stirring occasionally. Pour off the dirty solution and check the magnets. If they still feel like they aren't clean, or residue comes off on a clean rag or paper towel, remove the magnets and wipe the container. It wouldn't hurt to roll the magnets around in an old rag to remove some of the residue. Then put the magnets back and repeat.

I would tackle it like this. If some are painted and some aren't, and you want to get rid of the paint, start with those. Put those in a plastic paint storage container with a lid, cover them with paint stripper, and seal the container. Every so often, stir the contents so all the surfaces get exposed. You might need to let it sit overnight so all the paint dissolves.

There's a possibility that what's getting on your hands is some kind of grease or oil. Add the unpainted magnets to the container (if you decide not to strip the paint, start here with all of the magnets). Pour in enough 90+% isopropyl alcohol to just cover them, and stir everything well.

Add a squirt of liquid dishwashing soap, like Dawn, and stir everything well. Add some hot water (as hot as will come out of the faucet); roughly equal in volume to the alcohol and soap. Stir the magnets to get all of the surfaces exposed and to agitate them. Let it soak for awhile with regular agitation. Then pour off the dirty water and see of the magnets feel clean, with no more residue.

If not, repeat the soap and hot water. When they feel clean, fill the container with hot water, agitate to rinse everything, then pour off the water.

Dump the magnets onto an old towel, flatten them into a layer, then wrap the towel over them and roll everything around inside the towel to dry them all off. Then let the magnets air dry on a dry towel.

I'm revising this answer after further research.

Considerations

Neodymium magnets are made from an alloy of neodymium and other metals. The magnets are typically plated with nickel. Pure neodymium is very reactive, oxidizing quickly in air and reacting quickly with hot water (much slower in cold water). I don't know the extent to which the neodymium in the magnet alloy is similarly reactive (the plating might be for protection as well as appearance). It would be safer to just avoid cleaning with water.

Neodymium magnets also start to lose their magnetism at a relatively low temperature (below the boiling point of water), so don't expose them to temperatures above "warm".

If some of the magnets are painted, that might be a cheap replacement for nickel plating. If so, I wouldn't rush to remove it if it isn't chipping off. Neodymium magnets are typically made from small particles of the alloy, either using heat and pressure to fuse the particles or using a binder. The alloy is a silvery-gray color. If the magnet is not plated, it will have a grainy silvery appearance rather than looking polished.

If some paint is already chipping off and you can see a mirror-like silvery surface, the paint was added to plated magnets so it would be safe to remove. Even if the paint is just "decorative", removing it is messy. I'd be tempted to not remove the paint from any painted magnets that aren't chipping.

Cleaning

So there are two things to clean. You might want to remove the chipped paint for appearance and to prevent further chipping. What you describe as similar to getting newsprint ink on your hands may be a coating of grease or oil put on the magnets for protection. That could be grayish, but the magnet spheres could also act like a ball mill, grinding paint flakes to a fine dust, adding color to the grease or oil.

Paint removal

Put the magnets to be stripped in a plastic paint storage container with a lid, cover them with paint stripper, and seal the container. Every so often, stir the contents so all the surfaces get exposed. You might need to let it sit overnight so all the paint dissolves.

Degreasing

If you are already using paint thinner, that will also remove any oil or grease, and you typically buy that by the gallon, so you're likely to have some left.

You'll have three groups of magnets -- ones that have just been stripped and cleaned, painted ones that you didn't strip, and unpainted ones. The stripped magnets should already be clean. The unpainted magnets can be cleaned in paint thinner. Paint thinner will attack the remaining painted ones. Those (and the unpainted ones) can be cleaned with 90+% isopropyl alcohol.

Don't use the alcohol sold as a solvent in the paint department. That is denatured with other stuff that may attack paint that you want to preserve. Use alcohol (90% or higher) from a drug store or drug section of a department store (typically sold in pint or quart bottles). For the unpainted magnets, pick either solvent.

The cleaning procedure is the same for either solvent. Wipe out the container and put in the magnets to be cleaned. Pour in enough solvent to cover them, and stir everything well. Let it soak for a total of maybe an hour, stirring occasionally. Pour off the dirty solution and check the magnets. If they still feel like they aren't clean, remove the magnets and wipe the container. It wouldn't hurt to roll the magnets around in an old rag to remove some of the residue. Then put the magnets back and repeat.

When the magnets are clean, dump them onto an old towel, flatten them into a layer, then wrap the towel over them and roll everything around inside the towel to dry them all off. Then let the magnets air dry on a dry towel.