Refrigerating Hot Sauce is not required for it to keep safe to eat.
However if you do place the Hot Sauces in the refrigerator the Hot Sauce will keep fresher longer but you do not have to refrigerate Hot Sauce if you don't want too.
I've kept Hot Sauce out on the counter for years even after opening it and it's been okay but refrigerating the Hot Sauce once opened does keep it fresher tasting and makes it last longer.
Still Hot Sauce has lasted me for 7 months and longer when just kept out on the counter or on the table.