Oh, yes. They do. With a thousand parameters that need to be considered, but they do. Here is a collection of data from well known and trusted sources that support it. I wanted to collect all of them in a single place, since I usually search for this kind of data and it's hard to find everything in one single place. I hope you find this useful, add your stats in the comments if you wish.
Here's how social media influence buying decisions
Update 16.03.2010According to a research by Chadwick Martin Bailey and iModerate Research Technologies becoming a follower of a brand on Twitter or a fan on Facebook has a positive impact on the possibility to buy and to recommend a product, a service or a brand.

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First of all consumers trust real friends and peers the most, since recommendations from people known are the most trusted ones. A high degree of trust belongs to consumer opinions posted onlins (70%). Television (62%), Search Engine Result ads (41%) and Online Banner Ads (33%) rank way behind.

Source: Nielsen Global Online Consumer Survey 2009
It's very important to point out that blogs play an important role in purchase decisions: according to a US study 21% decide on a product or services based on blogs, 19% refine their choices, 19% get support and answers, 17% discover products and services.

Source: eMarketer, BuzzLogic "Harnessing the Power of Blogs", JupiterResearch - October 2008
Another (US) study points out the reasons for friending or following companies on social media: this happens mostly to learn about specials and sales (64%), to learn about new product, features and services (62%) and for entertainment (35%).

Source: eMarketer, MarketingSherpa and Survey Sampling "Popular Media Study" January 2010
According to another study by Razorfish here are some precise reasons why users follow brands on Twitter and purchase is an important one amongst them:

Source: eMarketer, Razorfish "FEED: Digital Brand Experience Study" November 2009
Another study, published on Harvard HBS Working Knowledge Blog on Cyworld data, points out that comments from friends on social media have positive effect in 40% of cases, neutral in 48% and negative only in 12%.
Last one, a (US) study by OTX about "The Impact of Social Media on Purchasing Behavior", that points out the attitute towards online recommendations. According to the study, opinions from consumer are valuable in 71% of cases, relevant in 68%, influence purchase decisions in 67%. Read the full study for more detail.

Source: OTX, The Impact of Social Media on Purchasing Behavior, 2008
Do you have more stats? Please add them in the comment and I'll aggregate them in the post.
Thanks, enjoy.