SSID Visibility to Neighbors

As far as a standard 802.11 (Wi-Fi) client is concerned, the same SSID means it's the same network. This is how you set up multi-AP wireless networks that allow roaming between APs; you set them all up to publish the same SSID, so the clients know those APs are all part of the same network and the clients can roam between them as needed. Some client implementations try to be smart about the potential problem you've raised (neighbors using non-unique default SSIDs), but you shouldn't assume that all clients will handle it gracefully and securely.

Having WPA2-PSK enabled on your AP will make sure that your neighbor's clients can't join your AP, but it won't keep them from trying (and possibly causing authentication error events on your router's system log). Having WPA2-PSK on your own AP won't necessarily keep your clients from trying to join your neighbor's same-SSID network. Your clients may prompt you for a new password for the network if they accidentally try to join your neighbor's AP, and the join fails because of the bad password.

If your neighbor turns off security on his network your clients may actually successfully join without prompting. That is to say, I've seen 802.11 clients that were susceptible to security downgrade attacks, where just because they knew a WPA2-PSK password for a given SSID, didn't mean they felt compelled to always require that network to use WPA2-PSK; if they saw the same SSID without security, they might just join it and not bother with security.