Yes, both wireless APs broadcast their respective SSIDs. It would only make sense for them to use different channels, although technically they would work using the same channel, but less efficiently. And the only non-overlapping channels on the 2.4GHz freq are 1, 6, and 11, so those are ideal.
As far as the SSID itself, using the same SSID does provide a pseudo form of roaming. I say “pseudo” because unlike a cellular network that would actually transfer you to another cell long before the connection was lost, your wireless client will NOT. It will cling to the current wireless AP until it’s lost, even if a stronger signal comes into range. It’s simply a limitation of wireless networking. It can’t preserve context from AP to AP, so rather than risk losing the current context, it holds it until it simply can’t any longer.