I might have another thing for people to try.
First, it is not true that you have to have more than one apple ID, we have had only one used over 2-3 phones for many years and apple has supported only one apple ID for them. We share contacts, notes, calendar, and a few pages/numbers files. Family share does not allow that (at least not at this time).
Every now and then we run into a problem where they don't work independently. Sometimes there is a new setting (like handoff) or something got changed after an IOP update.
We ran into this problem (call logs showing on both phones, not just the phone sending/receiving the calls) after updating to ios 9.3.4 this last week.
First call early this morning with Apple they checked that:
1) imessages and facetime were going only to the phone number of that phone (no other number, and not to any email address).
2) Handoff was turned off.
3) FYI, our icloud drive is turned on for both phones.
A couple of those had gotten set wrong (that person doesn't change their settings so not sure how that happened).
When checking this evening, sharing calls logs was still happening.
Just now rep had us log off of icloud (settings, icloud, sign off - at the bottom, choose to delete the notes off the phone - they are saved pm the cloud, but kept contacts and calendar on the phone). Then a reset (hold down home key & volume key). Then sign back on icloud (settings, icloud, merge contacts, calendars, etc.).
After the first phone finished, we did the same on the second phone.
Did a test (rep called each phone number) and they didn't show on the other call log.
All looks good (the calls made 30 minutes ago do not show up on the other phone).
So that is something to add to your list of things to try (especially tad44 who had the problem on two phones and not all 5, doing this on the two phones misbehaving will hopefully fix them up).
A side note, I always have apple-care on one of our phones (as problems with one typically are also on the other, even if it is a I can't figure out how to ...). When first level support can't fix it they pass me off to second level support and issues have always been fixed, including them giving me their contact number so I can contact them directly if the problem comes back. Well worth the price of the apple care (saves me time, frustration, etc.).