I just can't wrap my head around why the put-call ratio makes sense. Whenever there is a put buyer, there is a put seller, same goes for a call buyer/call seller. In other words, if there are a lot of call options for a given stock or index, there is an equivalent number of people who have sold those calls that are being traded. So the number of bulls is the same as the number of bears. For example, let's say there are 10 calls and 20 puts bought for the S&P resulting in a put-call ratio of 2 which should interpreted as bearish sentiment. But, equivalently, there are 10 call sellers (bears) and 20 put sellers (bulls). What am I missing here?
I think you are missing an important point regarding who initiates options positions.
We know that put options are more expensive than theory would indicate as discussed in Bondarenko (2014). Simply: put option buyers are especially motivated to initiate positions, more so than put sellers. Thus put open interest is a measure of put buyers initiating positions.
We cannot just look at the put option open interest, however; that might be larger or smaller just due to increased trading. Instead, we scale the number of puts by the number of calls. That corrects for fluctuations due to trading activity.
Given that put buyers are more likely to initiate a position than put sellers, the put-call ratio helps us estimate when put buyers have been more motivated as a fraction of the market.