I tried to find something on this on the USCIS website, but had no luck. There was a movement a few years ago to grant working status to "trailing spouses" from countries that would allow American trailing spouses to work, but apparently they had no success. In most countries these days, if someone on a work visa is allowed to bring spouse and children, the dependents are not allowed to work. There is normally a way to change the visa status (after some period of time), but it involves all the same qualification - which with the H1 category visas can be tough. (And, as they are discussing over in the Dubai section, involves certain risks if one or the other partner should lose their job. You aren't allowed to hold or apply for two different sorts of visas at the same time.)
Cheers,
Bev