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As far as I can tell, each country gets to choose the standards they require.
I'm only familiar with Austria and Switzerland. Both countries say that the test cannot be self conducted.
Edit: Worth noting that EU citizens don't require tests or vaccines to travel to other EU nations if their home country is rated green or orange on the new system. As it stands, UK citizens are excluded from that. Currently Austria and Switzerland forbid travel from the UK on the basis that it is a third nation.
Do the travel tests require you to do it in person and have your ID verified and have the swab done properly, or am I being hopelessly naive?