It might not be clear from the title without context that the setting for this attack is a local cache-timing attack, which can work cross-VM on the same physical hardware. So if you have an attacker running code on your physical machine, but not necessarily in the same process, account, or VM, you would be vulnerable to private key recovery.

Although the researchers describe mitigations that will dramatically reduce the information leakage, this is further evidence that it's quite dangerous to do software crypto on hardware that may be shared with an adversary.