Earlier quoted context omitted.
No, they don't. EDIT: I was wrong, this is one of the permissions they require on Android, however, not on iOS as Apple doesn't make it available to third-party developers.
On Android their app asks for "read call log" permissions. Luckily, I am running on root and can disable that access, but it's certainly possible.
It's a major privacy violation, with Google directly encouraging it for no good reason.
(Getting device ID should be a very rare, high-security permission; instead each app should get a unique app+deviceID. And seeing who is calling is likewise a rare thing that should be highly suspect. Or Google could add controls or do on-demand like MS and Apple do...)