I don't think there are any permissions you need to worry about unless you make enough money to report it on your taxes for a deduction. But I think the don't ask, don't tell policy works for most people teaching out of their homes.
My personal opinion about how easy it is... I think it's easier to go to their home to teach. There's something to be said about being on their turf that makes them more relaxed and open to discussion. But if you're doing more than one couple at a time or you have quite a few classes in a month, combining them and teaching in your home could be more convenient. I found that teaching outside of the home was worth the extra few bucks. I personally hated keeping my house immaculate, keeping my kids and husband out of the way, pets and allergy concerns, etc. The classes always start and end on time because I was never late! People coming to my home were always late.
True, you may put some miles on your car. You might spend more on gas (but if it's more than 20 miles away then I would ask for a small fee to cover the cost of gas. Clients are always cool about this.) But for me it was worth the trade off - no distractions, no cleaning, and I got to get out of the house for a few hours (and get a soy latte on the way!)
There are even some businesses that might let you use their space for a small fee or no fee at all since you are promoting their business in a way by using their space (like a yoga studio with prenatal yoga, a chiro clinic that works with pregnant women and infants, a shop that sells slings, breastfeeding supplies, cloth diapers, etc. Be picky!)