I've listed and sold several short sale properties and my experience is that offers are submitted to the first, and if there is one, to the second lien holder as well. Although sometimes, the listing or seller's agent submits the offer first to the 1st lien holder only and wait until they get an approval from them before submitting the short sale package to the 2nd so that they know for certain how much the 1st lender is willing to pay them. There is an estimated seller's HUD or closing statement accompanying the offer/s but for the 2nd loan, the pay off amount is very small, most of the time, less than 10% of the loan amount that the sellers owe. The 1st lien holder determines how much to pay off the 2nd lien holder. Now the 2nd lien holder may or may not approve of this amount and give you a counter. To clarify, the 2nd lien holder will also be paid off from your offer and that amount will be determined by the 1st lien holder.
Taking into consideration all the fees & charges the 1st lien holder has to pay off such as delinquent taxes (if applicable), part of the 2nd loan, listing & selling agent's commissions, property reports (Natural Hazards, 3-R for some cities), city & transfer taxes (depending on which city, some do not have transfer taxes or buyers pay for them), HOA transfer fees (for properties with HOAs), escrow charges, etc., the first lien holder may or may not approve an offer if it's too low and instead give what the acceptable price is or the "approved short sale price". This price may not be the amount of the offer submitted that's why sometimes, there are counter offers.
So the good news is, delinquent property taxes will also be paid off by the 1st lien holder at close of escrow so before title is transferred over to you as the new owner, it will be free and clear of any liens. The title & escrow company also makes sure of this that's why you have title insurance. And nowadays, we rarely see counter offers by banks anymore since they would rather sell it now than go through the foreclosure process.
I hope that helped. If you have any more questions, you can email me at email@example.com. Good luck on your purchase!