Generally, the only reason a drink or shot “on the house” is offered is because you’ve already been a pretty good guest. Whether you’ve been paying for your drinks as you go, and tipping well, or have just been polite and courteous to the bartender. In short, I assume if you are receiving free drinks, you’re not a jerk.
Therefore, I also am going to assume that if you’re not a jerk you would definitely tip a bit more than just the 20 percent of your final bill. Getting a drink or two gratis is always nice, and it’s even nicer to repay the gesture by increasing the tip a bit as a thank you. It will also help ensure you will receive the same treatment when you revisit the bar in the future.
If you are one of those people out there who has never received a drink “on the house,” you might want to question if it’s a you problem. Treat your service staff well, and they will treat you well in return.