I've sold many websites and here's what you need to know...
First: Make sure the owner actually owns the domain name(s), and has the authority and access to transfer them to you upon closing.
Second: Close the sale in person, and upon closing the sale make sure the seller allows you to access the "Administration Account" of the server where the site is hosted, change all passwords. If the site is hosted on a shared server you should get access to the Administration Level that manages the actual hosting of the site, as well as any other password protected Admin areas. Btw, I wouldn't buy a website that isn't hosted on it's own dedicated server, there's just too many ways for fraudulent activity to take place. If you have concerns about fraud you can immediately move the site after closing to a separate server system.
Third: Make sure the data, graphics & content is not plagiarized or copied. This can cost a lot of money later in legal fees. Very important!
Fourth: Make sure all backups of the site are handed over during the closing and make sure your contract stipulates how training and support will be handled with regard to running the site and backing up the site, including restoring the site if it goes down.
Fifth: If you don't have a good template contract you should hire an attorney who understands this process. It is easy to lose thousands of dollars and get literally nothing for it, if you don't have a professional walk you through it.
Sixth: There's a lot more to consider, so be smart and talk to someone who knows how to consummate one of these highly specialized sales.
I have great template contracts and I sell a bunch of website businesses. Our sales go smooth, but I have seen other Brokers and Attorneys make some very expensive mistakes.
I'm selling 4 website businesses right now. Two for others and 2 that I own. If you need help simply pick up the phone and call me at (214) 295-6364, Ext. 13 and I'll help you get through it unscathed... That goes for you attorneys as well.

Web reference: www.stallionassets.com

I'd be very cautious when buying an operating website. You're likely really buying nothing more than content and goodwill. The content will change quickly when the existing owner exits stage left and the goodwill may follow them.
Would you need an attorney? No, but it's obviously recommended that you make use of one. An astonishing percentage of attorneys are really grossly unqualified to handle even basic tasks, so I would suggest that you drive past your corner lawyer's office and use someone who has mergers and acquisition experience.

Definitely an attorney to review the contract but more importantly you need someone who knows about websites to ensure you don't get scammed. I'm a partner at a national firm, but I have bought and sold 3 of my own website businesses totaling $5 million. There are very few attorneys that understand analytics, Black hat seo, ppc, cpa, meta tags, and other crucial items. If you need help we can represent you and ensure your site will produce. Contact me directly for help...jlindsey@capiatlbbw.com

Web reference: www.capitalbbw.com

You would be best to work with an Attorney on the closing, but even more so on WRITING the Contract. Escrow is held by a Broker, Attorney or Title Company. Exactly how it works is up to each state law and the terms of the Contract - which is one of the many reasons you need an Attorney.
Precautions? It depends on the Web Site, exactly what you are buying, the nature of the revenue streams, what liabilities the Seller may have (if any) and what he/she is expecting, in the way of Contract Terms. It is not stuff that can be easily answered without knowing a LOT of detail. Once again, an Attorney at this point is crucial, as well as a CPA that handles business exchanges.

Web reference: www.bafgroup.com