It's not that Protestants think of themselves as "popes", rather, they believe in the concept of the "priesthood of all believers", which Catholics believe in too when they ask the question "Who is qualified to baptise" (for example).
All Christians are enjoined to pray together and discuss the meaning of the Scriptures together. It's not like we are all a bunch of imams out there workout out our own individualized interpretation of the Scriptures and their application to real life.
". . .workout out our own individualized interpretation of the Scriptures and their application to real life."
To Khomiakov that would have sounded a lot like 'every man his own Pope'. It sounds a lot that way to me.
We Orthodox don't work out our own interpretations--we accept the Apostolic Faith as it has been handed down by the Church from the time of the Apostles.
There is an amusing story from my old priest (now the Provost of St. Vladimir's Seminar) illustrated this: he and a number of other Orthodox priest were in a class for the Pittsburg Seminary's Orthodox D.Min., and the protestant faculty member teaching one of the courses proposed as an exercise that each member of the class (all Orthodox priests) write out their 'own personal creed'. The assembled all took out paper, and dutifully wrote, each one writing out the Nicean Creed (the original--no filioque), so that except for variations of quick v. living and spoke v. spake the texts turned in were all word-for-word identical.