GPS receivers
do not communicate with the satellites. All they do is listen to the satellite transmissions.
Most phones do do basic recording of your position (google call it your timeline), and you can access it from their web site
https://www.google.co.uk/maps/timeline. Google do allow you to share that information via location sharing.
All of the systems generally need some form of server to work as the tracker has to communicate back to something.
The electronics work is full of cheap and nasty which claim they do the job, but struggle to work reliably.
I doubt you would be able to build a system for much less than opentracking charge. One of the killers on cost will be the mobile contracts.
Basic checkpoint data as mentioned by Don could be done simpler and cheaper using Bluetooth LE tags, which you can get for ~£1/each