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How wise is it to collaborate with another professor while working as a staff under one professor? Is there a ethical conflict in doing so even if the other professor is from a slightly different area?
Also, how should one develop his/her own expertise while working as a postdoc under one professor? Is it wise to also work with other professor with slightly different research interest?
It depends totally on your status, and what exactly this collaboration entails. You say “working as a staff under one professor”. If you're on a short-term contract (say, post-doc), then it's probably not okay to do independent research on something completely different. You were hired on a project and it's unlikely that you have any significant time left for something else. On the other hand, if this new collaboration is related to your project and you involve your professor in it, it is okay (and he will welcome this initiative from which everyone involved benefits).
To the above, there are a few exceptions:
If you're working as an longer-term staff researcher in someone's group, then it is expected that they respect your academic freedom. To a reasonable extent, you are free to engage in new collaborations with other people, and not necessarily involve your group leader.
In contrast to F'x's opinion, I think collaboration with people outside your group is not just wise, but actually necessary, at least if you are working toward an academic career. You must develop both a wide professional network and broad independent research experience. As long as everything is out in the open, with everyone's knowledge and agreement, there is absolutely no ethical conflict.
This does not mean that you should shirk your paid job; any outside projects must be on your own time and must not interfere with your official duties. Similarly, you can't use your supervisor's resources without their explicit advance permission. And your supervisor may be happier if there's some chance of involving them in your outside projects. But ultimately, how you use your own time is your own business. (Any supervisor who tells you otherwise, or insists that you don't have "your own time", is simply abusing you. Run.)
If you are perfectly honest with all parties involved, then it shouldn't be a problem. Everyone involved knows where they stand.
One small point. In the UK, post-doc positions are funded by grant income. It would be strange if a post-doc had enough "free time" for other duties.
Let me give an answer from my own viewpoint - that of someone who hires and 'supervises' postdocs on a regular basis (in my field, mathematics).
I consider that a postdoc coming to my group comes for two reasons: (a) to bring new knowledge to our group, and (b) to learn things from our group. Both are good for the postdoc and good for the group - everybody profits from this arrangement. Learning things from our group is best done in close collaboration with the supervisor (me), while contributing to the group often benefits from a much wider scope.
So I tell my postdocs that I expect them to work half of their time on the project they are formally appointed for, and the other half on something of their own choosing - and I suggest that for this part they might also look for other connections within our group or department.
What this means for the OP is (as has already been said): discuss it with your supervisor, and ask yourself the question whether the 'outside' collaboration might also have benefits for your supervisor.
