Usually, there is only one vol surface (I have never seen or heard of anyone using two). Almost certainly the most advanced commercially available vol surfaces are built by voladynamics. They also offer only one surface for each underlying as well.
Even for interest rates, most of the time all quotes (ATM & OTM Swaptions, caps and floors etc.) are combined to one surface, as is outlined in this answer.
The reason one surface is preferred is that you need liquidity and the more data the better. Almost always, only OTM options are used because they are more liquid (exactly at the money hardly ever exists with price quoted options), which means you must have a blend of calls and puts if you want a vol surface for all moneyness levels (strikes).
There are lots of problems when trying to build equity vol surfaces. You can have stale prices, unreasonable small or large bid/ask spreads, the underlying asset and options on the asset can trade during different times on different exchanges, erratic prices, especially close to the opening and closing times of the trading session... Sorting all these difficulties out, still leaves a major problem. There is no consensus on how to model (cash) dividends, and you have borrow costs/funding costs. You can use a (continuous) dividend yield, cash dividends (implying that the observed stock price cannot follow geometric Brownian motion...), discrete proportional (discrete stock) dividends and so forth. Typically, a blend is used that uses different types for different maturities (due to lack of better data mainly).
Put another way, the problem is that the forward price is not directly quoted for listed options markets. Yet is an important quantity impacting option prices and implied volatility. Futures may be quoted but maturities frequently do not coincide with (all) option maturities. Interpolation is not trivial because future dividends (even times of payments) are usually unknown. Therefore, most practitioners (in my experience) use vanilla equity options to back out (implied) dividends. As stated in the question, the problem hereby is that put-call parity for American-Style options does not hold. Even for European-Style options, there are frequently issues with different trading times and erratic option prices etc. Since dividend payments are discrete in nature, you require a dividend schedule with dates and amounts to derive an implied dividend curve. This curve is usually noisier than one would hope for and commonly smoothed (via Kalman filter of the like).
Ignoring all these problems, let's focus on de-americanization. Frequently, a binomial tree is used as explained by Calibration to American options: numerical investigation of the de-Americanization method.
An elegant way that pins down to having qualitative dividend data (a dividend schedule consisting of declared dividends with dates and amounts and forecasts afterwards). This method is for example used by Bloomberg's BVOL vol surface. If you have access to BBG, you can have a look at the white paper which is 40 pages and outlines some of the complexities quite well.
Estimate the forward using spot, interest rates, dividends (forecasts) and ideally borrow costs/funding costs.
Use a local volatility model consistent with options of all strikes at this maturity. Alternatively, and a lot faster and simpler, compute the Black implied volatilities from put and call prices by using the estimated forward (implied forward in future iterations as explained below). This usually requires a PDE solver (or you use a binomial/ trinomial tree again). Lastly, use the estimated forward and the
implied volatilities in the Black model to compute the corresponding European option prices.
Imply the forward and dividends using the two nearest strikes on either side of the estimated forward from step 1 (make sure you quotes are reasonable for both puts and calls). The implied forward is the average of the forward obtained at each of the two strikes by applying put-call parity to the European option mid prices computed in 2), and back out a corresponding implied dividend.
Evaluate the results:
- Does the implied forward lie in between the two strikes? This is a minimum requirement.
- How different is the forward from the estimated forwarded (or implied if several steps are needed)? You will need some criteria to determine the optimal stopping.
- If the result in step 4) is not satisfying, use the current implied forward in step 2) and iterate until you are satisfied with the result.
These steps will give you Pseudo European option prices, option implied forwards and option implied dividends that are consistent with the observed American prices.
The calibration of the actual vol surface should result in the chosen (Pseudo) European option prices (only OTM, sufficiently liquid...) at any maturity to match the prices from the vol surface.
If you have access Bloomberg, you can play around with
OVME. The former will be the vol surface, the later an OTC pricer. Provided you look at a liquid option, and use the same snapshots of time, you can check how well these fit (OTC price from OVDV) the quoted prices from the exchange. Also, if you load a listed option in OVME, it will back out the implied vol (via a PDE solver in the case of American options).
Ultimately, there are plenty of challenges in building (equity) vol surfaces. Unless you have a large team of experienced people, it’s almost certainly best to simply use commercially available tools (voladynamics, Bloomberg, …). Companies like BBG will certainly not offer top of the class tools but will provide you with a complete solution from market data (option prices, underlying prices, qualitative interest rate curves, dividend (forecast) data, …) to pricing tools and portfolio tools. For example, if your team uses Kondor+ you will still need to get all market data, including vol surfaces from somewhere else to use this tool. This almost always poses problems with inconsistencies between market data (different conventions / methods used for different data).
This excellent answer by @Brian B explains why the pricing may not match after you computed them from the vol surface. In a nutshell, it is very crucial to model your borrow costs / funding costs and /or dividends properly. Timothy Klassen wrote a paper about this topic.