Many iPhone users started complaining lately about this popup message regarding "Message Logging" that looks like it's appearing out of thin air, sometimes when writing a message, but not only. The dialog box have a text input field and it states "Please describe the problem briefly". It looks like it is not necessarily related with the iOS version installed on your iPhone.
Chances are that when this message box pops up on the iPhone, stating "Messages Logging" one might go "Wait, what? Why would someone log my messages?" and start looking for a solution and luckily there is, however at this point no one is really sure what is generating this strange message.
This message is supposedly generated if you are using a Beta Profile OR if using a VPN (Virtual Private Network) connection. There is also a possibility that a keyboard App might generate the message. But no one is 100% sure at this time.
Delete Beta Profile is used
If the Message Logging prompt is generated by a Beta Profile, you need to go to Settings and than go to General. Now, scroll for the Profile label. Here you can click (tap) on it and you'll have listed all the available profiles. You can now select any profile and a "Delete Profile" option will be available. At this point you can simple erase the profile and reboot your iPhone after doing this.
If the Beta Profile solution does not work and you are using a VPN (Virtual Private Network), you need to uninstall the VPN to stop Messages Logging popup appearing again. I know there's a reason you have installed a VPN and if you insist on having it installed, you may look for another VPN solution or try to reinstall your proffered service.
To uninstall the VPN from your iPhone, go to Settings and than go to General. Now, scroll for the VPN label. At this point look for the "i" icon which is displayed next to the configuration label. A new page will open and you can scroll to Delete VPN option. Once you tap on it your VPN profile will be deleted. Reboot your phone and it should be OK at this point.
Uninstall External Keyboard
Say you are not having a Beta Profile nor a VPN service installed, or you might have performed the operations mentioned above yet still the Messages Logging prompt keeps popping up, there might be an external keyboard App that is causing the issue. You might say now: Really? And I have uninstalled my Beta Profile and VPN for nothing? OK, you might try to do this first than, however, there are very rare reported cases of the issue generated the a keyboard App. It looks like the error cannot be reproduced by only installing a keyboard App, however, it might be that such App in conjunction with one or both the above cases can have something to do with this strange message. To be safe, just uninstall any external keyboard you might use and reboot your iPhone again.
You should be fine at this point but please let us know if the above solutions work for you or if you have found any other solutions by yourself.
P.S. There are no reports that a Jailbroken device have anything to do with this message.