This is absolutely possible with the default Events Module.

Simply create a role (eg, Calendar) that should be able to add/edit events, assign the users to that role, and grant that role permission to Edit Module in the module settings.  Grant View permissions to All Users.

The drawback is that anyone in the Calendar role can edit/delete ANY event in that calendar.

If you want to restrict the users from editing events of others, you have two options that I'm aware of:

1.  Create individual calendars for each user and only grant edit permission for that user.  Use the Sub-Calendar functionality to "roll up" the calendars into a Master Calendar.

2.  Purchase XMod and the XMod Calendar add-on.  XMod does have settings that can restrict users to only editing/deleting their own events.