I've seen the "Free Public Wi-Fi" show up in many areas also. Usually this always shows up as an Ad hoc (computer-to-computer) network, which is the give-away that this is not an access point.
With the price of public Wi-Fi, I bought a Mi-Fi, which is a battery operated 3G/HSPA wireless router about the size of a flip-phone. For the price of an hour's usage at most public Wi-Fi points in the UK and Ireland (especially at an airport), I can get a whole month access on it, it works anywhere there is 3G coverage and the Internet connection is faster than what I've experienced with the Pay access points. Unlike public Wi-FI, it uses WPA2 encryption, so other users can't join the access point to eavesdrop on it.
A few tips I've learned when using public Wi-FI:
Most hotels and many restaurants do have free Wi-Fi. If you're unsure about a network, bring up the network list and ask the reception or service desk, as they should be able to identify their own network. In fact, when booking a hotel, I would rather choose a slightly more expensive hotel to have free Wi-Fi than one lacking it, as I've seen hotels that charge upwards of â‚¬8/hour for access.
For airports and shopping malls, expect to pay plenty for Internet access. On the other hand, I have seen free Wi-Fi from time to time. For example, the last time I was at Denver International airport, they had free Wi-Fi (can't remember the network name), which was ad-sponsored. The initial page would display a video advertisement. Once the ad finished playing, I could browse away after that.
If the network shows up as a "computer-to-computer" network, don't touch it