At my particular institution, I have known several who have transitioned (all older than me, and most who had done postdoc first). Basically what is required to transition is at least one of, preferably both of: A) 1-3 top-tier first-author papers (Nature/Science), or B) getting several small grants or one large grant adding up to at least 150K/yr. Politics can also play a role, obviously.
But I don't know for sure yet whether I will want to transition. No question, tenure-track is more prestige and pay, and research associate/staff scientist was originally created as a position for people who were too old to be postdocs and not willing or able to become faculty. I think the position is changing somewhat to be simply a "middle ground" between postdoc and faculty, less dependent on age.
But I joined this profession to do great research, not write grants all day, and I am given a great deal of freedom to do that, so I'm not seeing right now much benefit in becoming faculty other than pay. Maybe my perspective on the importance of pay will change if I have a family.
So, to answer the question directly, right now I am thinking I will become a PI iff I have to do that to push forward my research adequately. If, on the other hand, I can find a PI who is willing to do all the boring grant-writing work, pay me, and give me a lot of freedom, maybe I won't. Right now I have such a PI, which is awesome. After being this spoiled, I could never settle for the normal "you will do these experiments, serf" relationship a lot of PIs have with their underlings.
Someone else's grant funds my paycheck, but in practice, staff scientist is often a position which is intermediate in freedom between postdocs (who virtually always pursue someone else's research goals) and faculty (who, in theory, always pursue their own, once they're done writing grants and doing paperwork). So, I work "under" no one, but I do have someone who expects me to spend roughly 30% of my time on their research goals, and people would refer to me as a member of the X lab (where Dr. X is the faculty member paying my salary).
Staff scientist typically connotes a relatively senior, but non-tenure track position. So, someone who either did not want, or could not cut it in, the cutthroat world of tenure-track positions. The hours often are lighter than for professors. It is typically filled by someone in their late 30s or early 40s and who has done a postdoc.
In my case, and at my institution, it is different. I got this position because I and the person hiring me felt I deserved better salary, benefits, and prestige than a postdoc, whose salaries are fairly rigidly set by the NIH in the mid 40Ks. I can still become a professor later (the main way to do that would be to successfully write some grants). So overall, for me, this position is a step up from the likely alternative of postdoc, but for someone in their 40s whose competing alternative is faculty member, it would not be nearly so appealing.
A related position, slightly higher up the food chain but more definitively non tenure-track, is that of "core director". That means you run a "core" (a facility that all the researchers at an institution can use for a particular kind of experiment or analysis). A bioinformatics "core director" would then take datasets from researchers who come to them and perform analyses on them, and by playing their cards right, would get lots and lots of coauthorships.
Basically, you have to play it smart, find a niche, and you can beat the averages.