Do you ever see the person you escaped Helgen with again?

Yes. Depending on who you escaped with, you will see them again either by joining the Stormcloadk Rebellion, or by joining the Imperial Legion. However, even if you didn't follow the other guy, he will still be in the game. [SPOILER]:

When you follow Ralof in the game, he will be at Gerdur's in Riverwood. However if you joined the Legion and started the Jagged Crown, Hadvar will be at Korvanjund where you find the Crown and Ralof will be removed when you do the Legion quests. Everything else is the opposite when you follow Hadvar and join the Stormcloak Rebellion and go to Korvanjund.