1. The most common and oldest pronunciation is through curling and pressing the tip of the tongue against the upper teeth. The way it is pronounced with a Russian accent and pronounced by many sefardim, Chabad, and chasidim.
2. Within the last few centuries what is known as, the brisker Reish, its pronounced from the back of the throat. Its most common among modern Hebrew speakers and is of French and German origin. The sound of it is, that of one gargling water in back of his throat. It is pronounced that way in a few European countries. The Yerushalmis who came over from Europe spoke that way although it is a modern version of the authentic Reish.
3. The American R sound which is a liquid sound made from the lips and the outside edges of your tongue against the inner edges of the rear portions of the gums, next to the rear molars. It is similar to the W sound which why kids mix it up.
Now Your wondering which is the correct or authentic way to pronounce it. Well there is no way to know that as there has always been different accents based on languages. However the ספר יצירה (Sefer Yetzirah Perek 4) when discussing the 5 Sounds of the mouth it Groups Reish amongst the letters that emanate from closing of the Teeth. Which are Zayin Shin Samach Reish Tzadi ז ש ס ר צ. This appears to be as the First way it is pronounced through the closing of the teeth and the tip of the tongue. However in other places the Reish is grouped with the Seven Osiyos that are Doubled as soft or with a Dagesh known as בגד כפרת . Which correlate to the Seven days of the creation. The Reish would have to than, be coming from the throat or other area if it were to be pronounced softly. Which could be either the the Brisker Reish or like the English R sound.