It would seem like the starter would only follow you around up to Mr. Pokemon's house. Someone elsewhere speculated that since the starter Pokemon isn't the player's yet until after returning to New Bark Town after the first rival battle, it would remain outside of the Pokeball just to "accompany" the player. Elm gives the player the starter just for the Mr. Pokemon errand originally
So, perhaps now in HG/SS, the starter just follows the player up to Mr. Pokemon's house to show that it's not quite the player's yet. Then, once the Pokemon becomes the player's after the first rival battle, it goes into the Pokeball because it's finally the player's. Maybe? o_O
I dunno. But it just seems kind of unlikely that the starter will follow the player around throughout the game. (Which is kind of disappointing.)