Do countries that implement the death penalty only do it to their own citizens?

The US, despite what some have said here, has absolutely executed foreign nationals, although it isn't common. The US is part of the Vienna Convention on Consular Relation, but it hasn't always abided by the rules.

The Supreme Court has permitted numerous executions to go forward, saying that federal courts lack the power to address the issue if the lawyer appointed to represent the prisoner failed to timely raise it in the state courts. At least 34 foreign nationals have been executed in the modern era of the U.S. death penalty.