When you enter a new country do they report your entry to your home country? Do countries track where their citizens are?
There are currently 7376850 different combinations of citizen from "country A" travels from "country B" to "country C" so it's almost impossible to give a comprehensive answer.
However, in most cases the answer is "no". If you are a US citizen, crossing the border from, say, Denmark into Sweden, neither Danish or Swedish authority would report anything to the US authorities.
Countries typically keep track when you arrive and when you leave, but exchange of data between different countries is fairly rare.