It depends on the version of OpenSSL and yes ANY device may be in danger.
And no. It is not only a server side problem. If OWB is using the broken SSL port, which is possible with the OS4 Port of V1.23, and accessing a compromised server, the server can read out 64KB of memory on your computer, which for sure contains plain keys. MorphOS OSB V1.23 at least is safe. You need to ask Kas1e what version is used in his port.
Even if it is a known site with already got updated, you do not know if it was compromised before and still spys on you.
Same for any application using SSL. Check the versions used.
You need to update or any kind of server (mail, https, ...) may spy on you.
Geit