Theoretically, this is a more difficult problem than it looks like at first glance. Unfortunately, existing literature taking into account a proper dividend consideration is rare (at least from a practical viewpoint). There are several options:
1) Use what is called "De-Americanization": In this case, based on your input dividends (maybe based on other sources like dividend futures or synthetics), you price the American options with a flat vol (by properly taking into account the early exercise features and dividends). This could happen in a tree for example. Hence, you are deriving the flat vol, which prices the market quotes of American options. Obviously, you somehow treat American options as European ones in this case but it works amazingly well (except in very special cases, which I will mention below). I know that this is a very popular approach in market practice. Once you have calibrated this "European" implied vol surface you can go on, e.g., with your local vol calculations to price exotics.
EDIT: Based on the comments, a bit more detail (but please see the below paper for the full details): Given a fixed listed expiry T and a set of N American options with different strikes, you are fitting a binomial tree for each of these options and back out the implied vol (or, equivalently, the "up" parameter u).
2) Alternatively, you can start with a parameterization of a "European implied vol surface" (i.e. you have model parameters per time slice describing the skew). Then you are directly fitting these model parameters, e.g, in the local volatility model to market quotes of American options (by using Dupire equation along with some finite difference scheme). This is definitely more sophisticated than the option before but obviously also numerically more demanding
3) In step 2, you can also simultaneously calibrate dividends. At least in theory. In this case, you are calibrating your model parameters and your dividend payments simultaneously to the market quotes (with some optimization technique). This again is even more challenging and practical evidence is questionable since you have to deal with potential overparameterization. Note that extracting dividends or setting a reasonable range for them is very difficult due to illiquidity of dividend derivatives
Market practice tends to favor option 1. But now, here is the challenge:
Suppose, the dividend yield for your underlying is high, interest rates are negative and volatility levels are high (maybe even strongly inverse). There are plenty of such examples currently in the market. If you go through the math and calibrate put and call options separately, you will see that the implied vols for puts and calls with the same strike may differ significantly (due to early exercise value being significant in the above case). Then the question is: Are market participants (especially market makers) maintaining two different implied volatility surfaces for one and the same underlying? One surface for calls and puts respectively? I don't know the answer but I can provide many examples where numerical results strongly advice this.
Hope this helps in providing you some insight into market practice and challenges
EDIT: You can have a look at the following paper for details of the "De-Americanization" approach: