I finally got this thing to work! I have an iPhone 6+ and a 2014 Hyundai Sonata Limited. I tried most of what everyone was recommending and nothing seemed to work, even including resetting All Settings.
What worked for me is this:
I have 4 iPhones (6+, 5S, 5, and 5C) and they WERE sharing one Apple ID. The 6+ and 5S were on iOS 8.02 and iOS 8.0 respectively. The 5 and 5C are still on iOS 7. I had to create a new Apple ID for my son's 5S and set his phone up as a Family Share on my Apple ID, since my Apple ID is the "Organizer". Then I had to ensure that the Share My Location settings were correctly setup, which means that the proper device was selected in the "From" box.
Here is how you do that:
Select Settings.
Select Privacy.
Select Location Services.
Select Share My Location.
Select From.
Then make sure that you select the current iPhone that you are using. (In my case it was my iPhone 6+)
Select Back.
Now in the "From" box it should show "This Device". (I did this on both the 5S and 6+.)
What was originally happening when my son's 5S was on my Apple ID, both his 5S and my 6+ were showing on the "From" selection page on both iPhones, and that was causing problems, apparently.
I also have "Handoff" turned off as well, since I don't have any Macs or iPads to use for Handoff.
Here is how you find that setting:
Select Settings.
Select General.
Select Handoff & Suggested Apps.
Turn off or on Handoff. (Mine is off.)
After all of that was done, I deleted my Sonata from my iPhone Bluetooth Settings and deleted the pairing in my Sonata. I restarted the iPhone 6+, paired it again with my Sonata, made a few test calls, restarted the iPhone, and made another test call. It finally works!! I will keep testing it this weekend to ensure that it is a good fix.
I really can't explain why this worked for me, but I believe there are software glitches with the new iOS 8 Location Sharing and Handoff features, especially when the devices share the same Apple ID.