Give HP as in additional MAX HP? Or heal HP as in give HP when you're not full HP? I tested the weapon, and it works as it should: it recovers HP when you use the basic attack.
I have tested it out again, it works now. I feel confused, but I guess I used it and dealt 0 dmg, or tried using skills as well that did dmg, but did not use drain attack by then to heal. I was sure I used drain attack and dealt some dmg and not recover HP at a point, but it works as it should so I guess I thought wrong. Sorry for the wrong report.