Do you think staff should be on the lookout for rules violations, should they do the same browsing that they did as regular members and take care of any rules violations as they see them, or, do you think staff should just wait for a post to be reported and then take care of it? What do you think?
Do the same browsing they did as regular members. Their task is to be involved also in the community. and it's also their job to take care of the received reports from other members.
It depends on the nature of your forum but if there are "trouble spots" it is reasonable to expect them to glance through the section from time to time.
The feature for reporting posts is extremely useful, but you can't expect members to always alert you to spam and other rule violations. Surfing the forums that attract the most trouble is a good idea, but never just leave a place unattended and expect your members to report rule breakers.
I like my staff on forums to be proactive, and get ahead of problems if they can. But, at the same time, most staff on forums are volunteer. That means they may not have the time to dedicate to the forum that they need to be proactive. So, I don't mind waiting for a problem to be reported and then someone dealing with it through the normal moderation channels.
What would the point be in acting just on reports? If spam (or another issue) is staring in the face, then it should be dealt with there and then. Much better to have moderators who participate. As MrJohn mentioned though, moderators are doing it voluntarily, so try to find ones who are also interested in some level of participation. Also give an incentive such as premium membership, if your forum has membership levels and it's genuinely useful (for example, DP's includes additional functionality for the keyword tracker and access to custom products, among other things).
If your moderators sit back and wait for threads/posts to get reported, your forum will turn into a spam infested forum with ton's of low quality posters and rule breakers. Not all members will report violations because it's not their job to do so. It's a forum feature available to members if they wish to use it. It's the mods duties to review the threads/posts in their assigned sections when they online handling things accordingly. I'm a moderator on a webmaster forum and I read every single new post on the forum everyday and night. I find things other moderators missed but a good moderating team can keep a forum active with quality posters and less spammers/rule breaker's. If members get the impression the forum has a poor moderating crew, people will join to do sneaky things and before you know it, the forum lost all its loyal quality posters and has a bad reputation.