Such a person is called a 'frontier worker' (images of racoon-pelt hats and Bowie knives come to mind!):
A frontier worker is defined as any employed or self-employed person who works in the territory of a Member State and lives in the territory of another Member State to which s/he returns as a rule daily or at least once a week.
If such a worker becomes partially unemployed, the state in which that person is normally employed is responsible for paying jobseeker's benefit. However, where a frontier worker becomes fully unemployed, the state in which s/he is resident will be responsible for paying the benefit.
A frontier worker who resides in Ireland does not require any Irish contribution to qualify under Irish legislation if s/he becomes fully unemployed.