<<<If you shorted American option, would it be like you suddenly have to sell (call) or buy (put) it when the buyer exercised its right?>>>
For a call option there is only once circumstance where it is the holder's financial best interest to exercise the option early, just before the ex-date for a substancial dividend particularly when the extrinsic value of the option is less than dividend amount. That does not mean that is the only time a call option will be exercised early, but it is quite unlikely at any other time.
For a put option the situation is somewhat different because of the carrying cost associated with put options. To hedge a long put option with stock, you have to buy shares of the stock. That means the owner of a hedged put option will own both the put option and shares of the stock. The money he has invested in the option and stock is not earning interest, so the interest he is losing is called his carrying cost. With a European-style option the cost of the option can drop below its intrinsic value since it cannot be exercised early. With an American-style option the cost cannot drop below its intrinsic value since it could be exercised immediately for a risk-free profit if it did.
If the bid quote for an American-style put option is less than the intrinsic value of the option, there is a substantial risk that anyone short that option will be assigned early. If the bid quote for an American-style put option is more than the intrinsic value of the option, there is almost not chance of an early assignment.
<<<But there are thousands of buyers and sellers in the option market. So I wouldn't have any idea who is the buyer of my option even if I short sold it, would I?>>>
The buyer is the Option Clearinghouse Corporation (OCC) any time you sell an option. Similarly, the seller is the OCC any time you buy an option.
Say you sold 5 contracts of a particular option, your friend Harry bought 5 contracts of the same option on the same day, and the total volume for the day was 5 contracts. That means the volume represents the 5 contracts you sold and Harry bought, but it does not mean Harry bought the contracts from you. There is no link between you and Harry as far as the OCC is concerned. The next day, if Harry exercised the options, the OCC would randomly pick someone who is short 5 contracts and send them an assignment notice. It might be you, but it is more likely to be someone else.
I hope that answers your follow-up question to Leo Enoch.
<<<Please tel me what it's like to short American options.>>>
Trading American-style options is not that different than trading Europtean-style. It is true you always have to be prepared for an early assignment when you are short, but it is not that common. If you understand why someone might exercise early, you know how much risk there is.
Options books I have read, discussions with options professionals, experience.