This app was mentioned in 2 comments, with an average of 2.50 upvotes
I'm using a Blackberry, but there is an Android version of the app I use which my SO uses. I don't know which permissions exactly the app uses, but on Blackberry it is location (obvious), shared files (for exporting data) and camera (for controlling the flashlight, basically useless). It works just fine without the extra permissions other than location services, and online synchronisation is optional. But as said, this is my experience on Blackberry, I'm not 100% shure about Android or iOS.
The authors seem to be somewhat privacy aware (scroll down to "Using the mobile app"), if it helps you for your decision to try it.
Have a look at MeeRun (https://play.google.com/store/apps/details?id=com.bwa.meerun&hl=en) I use their BB10 app, and while I do upload my data, they have the option to not upload data (it is new feature for them). They have resisted hooking into other sites in the past, for privacy reasons.
They are pro privacy - have a read through their privacy statement :) https://www.meerun.com/#!privacy it is detailed.