I am also having this problem since 3 weeks ago. My missed calls/call logs are showing up on my younger son's phone and vise versa, but not on my wife's and older son's phone. What I've found is only my phone and my younger son's phone are recently updated to ios 9.1, other two phones are still at ios 6 and ios 7 which don't have this issue at all. So i think Apple needs to come out something to to fix it! I have been using iphones since the 1st day it comes out! Never had any problem like this before even we are on the same Apple ID. For now, just about miss calls/call logs issues, how is about if your private messages are shown on other unexpected devices?...really annoying!
What i have done/changed on my phone's setting are:
-Restore iPhone in DFU mode, completely fresh installed: No help
-In iCloud, turn off everything, except Find My iphone ON, Share my Location OFF: No help
-Setting->Phone->Calls on Other Devices OFF: No help
-iMessage OFF, FaceTime OFF: No Help
-Setting->General->Handoff & Suggested Apps OFF: No help
-Sign Out iCloud/iTunes & App Store completely? No try yet, what a point? If so, rather go for Android!
My conclusion, I feel regret after upgrading it to ios 9.1
Thanks,