@Wolfgang_S is correct, the openHAB cloud is just an intermediary between openHAB and Google Assistant.
Before you go down the private-cloud route, what do you want to do with IFTTT? If you want to use IFTTT to trigger actions in openHAB, you’ll need the private cloud or one of the somewhat ugly workarounds (e.g. Tasker). However, if you only want to have openHAB trigger IFTTT actions, you can easily send commands using webhooks.