I had a guy tell me earlier that most bass guys learn how to construct guitar chords then play arpeggios nearly the entire time. I've never even thought about it other than power chords. Makes sense kind of, but I'm just wondering if anybody "really" does this, or if you all think about scales, or what.
I'm a scale guy.
Now it's 1984
Knock knock at your front door
It's the suede denim secret police
They have come for your un-cool niece
I think about scales, but I wish I started thinking about chords first. Everyone seems to do scales-->chords, but in my mind, the natural progression should be the other way around. The chords are more important than the scales and it's much easier to add in an extra note in the scale once you have the chord thing down than to revert from scales back to chords.
I take the chords that my guitarist's use and I play around in them. Mix in some chromatic notes, some wonky intervals, and a cool rhythm. BAM. A sweet bassline.
I think of notes in terms of Intervals, which highlights chord tones. When someone is playing a Diminished chord, what notes do I feel need to stand out?

Think of Chords as scales, and scales as chords. Use each in unconventional ways, and have fun!
Back when I was taught guitar (when dinosaurs also roamed the earth, but I digress), we learned chords as notes completely unrelated to scales. For instance I knew an A minor to be A - C - E but really had no sense of how that related to the C major scale. We were taught to sight read like demons, but no sense of the theory.

To continue the points progbass and Bales have started, you really need both. If you work your arppegio patterns through a particular scale, you start getting a sense of the chords that work within that scale and how you can build bass lines around them. For instance if you take the first few intervals of C major you get:

Root- C E G (C major)
2nd D F A (D minor)
3rd E G B (G major)
4th F A C (F major)

and so on.....

And then you get to relate those to the modes, but that's a topic for another thread...