I don't know for other heads of state, but the French president has a (small) service dedicated to his protection, the Groupe de sécurité de la présidence de la République (Security Group of the President of the Republic), whose name is a little more explicit than Secret Service.

https://en.wikipedia.org/wiki/Groupe_de_s%C3%A9curit%C3%A9_d...

Fun story: at the time of its creation, in 1983, under president Mitterand, one (unofficial) mission was the protection of the illegitimate daughter of the president.