Is meeting people on the street the only way to make friends?

The friend function is unrelated to the streetpass plaza. One person on my friends list has only appeared once in streetpass plaza - which was from when we set it up to begin with.

Miis can come to the plaza from playing multiplayer with them in certain games, like Mario Kart 7. Note that these can not be used for the streetpass games.

Other than that, you need to meet them "on the street", as far as I can tell.

Another way to add Miis to the StreetPass-Plaza are special Miis. You can scan the QR-codes, to get the Miis and can send them to the plaza. They cannot be used for puzzles/find-mii though.

Resident evil special miis: http://www.nintendolife.com/news/2011/09/get_your_special_resident_evil_miis_now

Some other Miis: http://streetpassberlin.wordpress.com/2012/01/16/mii-mit-goldenen-hosen/

Scan the QR-codes with the Mii-Maker. That works directly from computer-screen. You can send them afterwards to the Mii-Plaza, but they are only at one place at once.

There are friend codes in the 3DS as well, you can access the friend utility from the main screen by tapping the orange smiley icon. You can add friends locally by connecting consoles via WiFi or by inputting their friend codes.
On the Wii, Miis roamed between friend consoles as well so I believe this will also be the case.