For both, Brits and Argies was common carried weapon the FN FAL, in Argentinas was produced under licence form FN Herstal )Belgium) at the Fabricaciones Militares Factory.
They had a Para version based on the original.
It's not uncommon practice in the special forces and elite forces to carry your own weapon, in fact , Brit Special forces carried M-16"s based on the info I have.
Argentine's special forces used to a very little extent, but mainly Navy commandos use the Sterling.