Consider a molecule of ethane, C2H6. Let's talk about the carbon atoms:
- They are each connected to a hydrogen atom.
This is true, but misleading because actually
- They are each connected to three hydrogen atoms.
There is no difference between "They are each" and "they each are"; web and book searches find that the former is much preferred.
Still talking about the carbon atoms:
- They are connected to each hydrogen atom.
Nothing wrong grammatically, but it's' not true. This means that the carbon atoms are connected to each and every one of the six hydrogen atoms, but each carbon is connected to only three of the hydrogen atoms. "Each hydrogen atoms" is wrong because "each" means "each one" and must modify a singular noun.
- We each have own own car. We each have our own cars.
The first means one car a piece. The second will probably be interpreted the same way (two of us times one car a piece equals two cars). But there is the possibility that one or both of you have more than one car. This makes the following sound a bit odd:
- We each have our own noses.