Mmm, this is were we need some facts again isn't it? I think you are both right: there is potential to make cost savings through bulk buying but in practice LAs were often bad at this and schools ended up paying over the odds. Since then, some headteachers have probably saved money, and some have fallen for sharp practice by IT sales people.
I think that with such large and complex organisations as schools, you cannot guess what system will be cheaper or more efficient, you need to see the evidence and look at the training given when functions are transferred between sectors, and at the checks and balances in the system (I suspect headteachers may not have had enough support and training on procurement when the shift started - and also that Labour was naive about its new private sector friends). And there will be pressures in both sectors to cut costs at the expense of quality, which is why we need good, effective regulation (I worry about this with the current government), transparency, accountability and all the rest of it.
All I have been saying is that these problems apply to both sectors. And both are clearly here to stay in the provision of public services, so if we want the best outcomes possible we need to look hard at what actually happens and why.