It would be better to say that happens in other countries but it doesn't here.

That happens in other countries,but it doesn't here ..but it doesn't happen here Either of both is correct.

It would be clearer, obviously if you repeated either the object or the verb .. but it doesn't here. .. but doesn't happen here. .. but it doesn't happen here. However, your sentence is just about ok. Others may disagree, but I think it's acceptable as it is.