I do both
I've long been an advocate of disc ownership, because you can watch what you like, when you like, and no one can control it or take it from you. That said, I now also have streaming subscriptions, because now that I live in Canada and Netflix doesn't suck, it's totally worth it. (Plus, I have the obligatory US connection to get the "proper" versions of Netflix and Hulu, which are both awesome.)
Being able to check out series I've heard of on a whim, legally, instantly, and with a surprisingly good indexing system is great. The recommendation system might be useless but most of my recommendations come from friends. Hulu's anime section is unbelievably good for a free service, and Netflix is well worth the $8 a month I pay for it.
That said, I do still buy discs, in two cases:
1) Something isn't available on streaming.
2) The content in question is something I really enjoy and want to have permanently available, especially as blu-ray.
So the library on my shelf is becoming a list of favourites in the highest available quality, and general watching is done via Netflix or Hulu. Honestly, I don't actually know anyone in Vancouver that even watches traditional TV any more.