In general, the pricing for a club soda at a bar is as follows.
The menu price of a club soda is usually $1-$2. Following menu conventions this is almost always assumed to be (unless explicitly stated otherwise) the price per drink.
If one walks into a bar, orders a single club soda and nothing else, this $1-$2 menu price is the price they will pay for their one club soda.
If however, after that first club soda is ordered, any subsequent club soda will usually be free of charge. This is both because the $1-$2 essentially covers the marginal cost of 4-8 club sodas and also because it is assumed that proximity to the bar (caused by the club soda in ones hand) will lead to greater expenditures by the club soda drinker on higher profit items such as well drinks or any variety of fried starch.
Note that in groups if one group member pays the menu price for their one club soda it does not imply that all future club sodas for the group will be free, only that future club sodas for that specific individual will be free.
If a person instead were to go into the bar and order food, then proceed to order order a club soda, the menu price of a club soda would be dispensed with immediately. The price of all club sodas including the first, would be zero.
Contained in the normal menu price for club soda is a high premium for being able to take up a seat at the bar. By ordering food one has essentially already paid this hidden "door charge" and there is no need to make a huge profit on the customers club soda. By giving away the club soda for free the bar also makes the customer feel more welcome and is likely to spend more on big ticket items like well drinks or more food. Such free drinks also increase the likely hood of future return business at very low cost.
The bartender, it should be noted, has no obligation to utilize this de facto pricing structure. If a customer has been particularly rude, the bartender may opt to charge them for one, several, or all of the club sodas consumed during the sitting. A repeat customer who rarely tips is less likely to receive any of the above discounts.