I write in both. Sometimes I feel like I have no inspiration with a computer, so I will write in a notebook to at least get started and then I switch to the computer. Poetry, I almost always write on paper and then transfer it.
i just write it on the specific website i am sharing it in, if i am writing something for deviantart, i write it here, or for another example if i write something on a XYZ website, i write it there.. helps with classifying ideas according to the audience of the website, ofc if the writing tool in that website doesn't sux hard that is; and the most important thing is to save it occasionally if it is a long or quality writing.
I type everything. I type first drafts, final drafts, notes, plans for future chapters, reminders about the current chapter, passages I want to insert at the right time. It all gets typed into either OpenOffice or Notepad.
These days I don't even use my paper note book while I am away from my computer as often as I used to. More and more of my random ideas get typed into my phone as a reminder instead of written in pen.
There's really no "should" on this matter, it just depends on what's most comfortable for you.
My tendency is to prefer pen and paper for drafting or planning, typing for finished drafts. More recently, in order to conserve paper, I've done more of the initial writing using Scrivener. But I'm a pretty tactile person, I like scribbling on pages, circling stuff, crossing stuff out, and as I've had to go back and re-plan the story I've been working on the last year, I've also gone back to pen and note paper.
Usually it just depends on where I am when I'm writing. A lot of what I've written has been written during classes in school, so any of them are written on paper. But if I have a computer or some sort of device I can type on handy and can use it at the time, then I prefer that. It's easier to fix mistakes and edit that way.
Then again, I have very little that is solely on one or the other. Almost everything I've done has been split between the two because I work on them at different times with different resources available.