There is a condition called eczema herpeticum, in which HSV infects areas with eczema. It is quite serious and can be difficult to treat. It is quite uncommon, but still something you need to be aware of. I don't know if taking suppressive treatment for herpes (acyclovir, valacyclovir, etc) helps prevent it, but I would think so. This is definitely something for you to discuss with a dermatologist, or whoever you see for the eczema problem.
However, as long as you don't have actual eczema herpeticum, your eczema itself will not spread herpes, and neither will sharing a bed. You won't infect anyone with HSV unless that particular area (your genital area or mouth, wherever your herpes is) is intimately exposed to another person, i.e. sexual contact or intimate kissing.
If you haven't been to your dermatolgist since you caught herpes, make an appointent soon to discuss all this. Best wishes and happy new year.