Not all banks are created equal. Some personal banks don't really have good business banking. I would look around for the best business bank and use them. ammoQ makes a good point about transfers and fees but I think that is secondary to a good business bank that meets your needs.
Aside from the good suggestions about looking for a bank that specializes in small businesses and transfering money between accounts, the more money you have at a given bank, the more power you have there.
We use the same small bank for both personal and business accounts. Everyone at that bank knows me by name. When I need a personal loan it's easy to get the loan at great rates because of my business relationship with the bank. When my business needs a corporate loan it's also easy.
FDIC insurance currently covers $250K. If you need more than that, split the money between banks.
Our company's bank offers free accounts for all founders at no cost (but totally independent from the company's account). Personally, I'm keeping my current personal account as well as the new one, just in case.
Surely, transferring money between accounts in the same bank is faster, but I'd also like to have some money totally "isolated" from the company.
In addition to all the good suggestions so far, I'd say that you should definitely look for a bank that is friendly towards small businesses and has good business banking products and services. Often times, you can get fee's waived if you have a good established relationship with the bank.
Once you've found a few banks your interested in, sit-down with a personal banker and have a conversation with them. Let them know that your shopping around and would like to see what they have to offer.
Once you've made your selection. Make sure you get some face-time in with the bank. Banks are risk-averse. Anything you can do to strengthen that relationship will lessen their risk aversion. Of course, if you are overdrawn on your account all the time, stalking your bank manager isn't going to make the situation any better.
Here are a few other factors I look for:
We have our personal and business accounts at the same bank. We can transfer funds between our business and personal accounts as needed. Also, we have an unsecured line of credit that we can transfer funds between our business and personal accounts.
This all works fine because we have a single-member LLC for our business entity. If we had partners or other types of co-owners to whom we were accountable, a lot of this flexibility could become a liability and we would probably consider separation of our personal and business banking relationships to be more important.
Different institutions sounds not like a good idea to me. Transfering money from business to personal takes longer and probably costs more fees; when your personal account is overdrawn and the bank doesn't see the money on the business account, they might become nervous;
overall, things might become more cumbersome.