Would you compete with Public Sector?

A twitter & LinkedIn colleague has a blog post he refers to quite often - Not All RFPs deserve a Proposal. Generally his points are based upon how poor the RFP is written, if there is an incumbent, etc plus it is written from a proposal writers point of view.

But being on the public sector side of things, I'd like an opinion. Generally trade agreements and procurement policy give an exemption if the public sector needs to contract with another public sector organization (ie essentially with itself/another branch, etc). There have been discussions as to whether public sector 'should' be forced to compete, but on the other hand, should the private sector be forced to compete against the public sector?

I've had two situations where the discussion came up:
1) Flu shots for public servants - the requirement was to provide flu shot clinics across the whole province, from a private sector nursing/care provider. What came up in discussion was whether the health authorities could provide this service? They provide the flu shot clinics for the general public, couldn't they also be paid to provide this service to the public servants? Essentially the decision came down that they could compete for the work if they wanted it, but being separate entities, the health authorities didn't have the time to co-ordinate efforts to respond to the RFP as to their ability to provide province-wide coverage.

2) Training programs for new trades - in this situation, the buying organization wanted to have a competition between public sector (universities, colleges) and private sector (independent training providers). The process resulted in responses from only the private sector and one college.

In the private sector, yes, parent companies look to buy internally from its subsidiaries, when it makes sense for the business 'overall', if not the business looks to outside vendors. So what do you think the public sector should be doing?? Do we have the resources(taxpayers money and staff) to have public sector respond to RFPs of other public sector organizations??? As a private sector firm, would YOU compete with the public sector??