Posting Logos on Your Homepage to Increase Trustworthiness
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If you have been linked to from authoritative sites would it be appropriate to name them on a "As Seen On" section on your homepage with their logo. Do you need permission to use their logo on your site?
A followup question: would doing this help with the Panda algorithm.
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Technically, yes, you should get permission. If you use it, and the organisation subsequently decides it would rather you didn't, they will usually send a Cease & Desist notice, and you will be given the opportunity to remove the logo without running into any legal issues.
You are, however, unlikely to encounter a problem if you genuinely have been cited by the organisation in question, unless it was in a negative context.
On the follow-up question, I don't think adding the logo of an authority site will make much/any difference to how a site is treated by Google's algorithm, however the quality link, increased traffic and PR benefit should certainly make up for it.
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Logos are usually Trademarked and do require permission to use. Most sites, won't mind you posting their logo (free advertisement) but they will likely check your site / validity. Recently, I had to ask Sony Online Entertainment to use their logo, they said no problem.
That said, I have seen hundreds of sites post logos for all sorts off stuff they are no way affiliated to or with. Depending on your market it you may have also noticed this. I say that to say, personally a logo means nothing to me when I evaluate whether or not I want to trust said site.
I have nothing to add about the Panda question, but I hope that helps with the first question.