With private sales, you save the broker's commission, which in Baden-Württemberg can still be pretty substantial - you're often looking at 3-6% of the purchase price. That's obviously tempting, but the real advantage of a broker isn't just about money. A broker knows the market, filters out offers that look dodgy from the start, and connects you with serious sellers. With private sales, it happens pretty quickly that you end up stuck with hidden defects, unclear debt arrangements, or legal issues that'll cost you many times over.
With direct purchases, you really need a solid property inspection - it's basically not optional, it should be standard. And solid legal advice often becomes necessary pretty fast, especially when it comes to questions about the land register, old encumbrances, or renovations. These costs combined can easily approach what you'd pay in broker commission, and you'll have to battle through it all yourself anyway.
Based on my experience, I'd lean toward going with a broker, especially if you don't happen to know someone you trust. In Stuttgart and the surrounding area, the market's tight anyway - a broker helps you not get lost among a hundred other interested buyers. If a private listing really catches your eye, at least do a thorough check and definitely get an appraiser out there.