Hi there,
Actually, announcement app is also available for groups. Since you didn't use pages, there is no reason to turn on that. By the way, your initial question is about pages, and that is why our conversation is pivoting towards pages.
Anyways, to set the announcement for all group's members, you can checkout this settings at: EasySocial > Groups > Categories > (Choose desired group category) > ACL > Announcements >
Allow Announcements From = Any members as showin in the attached screenshot. By turning this option on, all group's members will have the ability to create announcements.
Regarding your inquiry,
> Or may be it would be better to do this in Easydiscuss - but there seems also no such a feature implemented.... or?
EasyDiscuss is a questions and answers component which is not suitable for announcement type of action.
Hope this helps.
Thanks.