My question is about staffing firms. If one receives an additional offer besides the one the recruiter was able to get, during the "think about it phase", is it a good idea to tell the recruiter about the other offer? What would be a good tactic to increase the initial offer from the recruiter, in this situation? Would the recruiter dig into the company name and the offer details or probably take the candidates word for it and try to meet them in the middle?
Is it a good idea to tell the recruiter about the other offer?
Yes, it is a good idea, but only if you keep it to general terms and don't tell the recruiter the name of their competitor (or give them enough detail to find out).
Why a good idea? It is a great bargaining chip - they are no longer exclusive to you and have more leverage for negotiation (with both offers / companies). You are in a position to compare the offers and ask either company to match certain of the terms of the contract of the other.
Being in demand is a good thing and reflects well on you - for the hiring manager.
Why not give identifying information?
Because not all recruiters have scruples. I have heard of cases where the recruiter actively sabotaged the other offer (that wasn't by them) - it was in their interest to reduce the options to those that will end up giving them a commission.
It doesn't really matter if the recruiter believes you have another offer or not - as far as they are concerned, you are negotiating. That should be enough.
The only time I would share this information is to force a company to make an offer when they have been dragging their feet. When doing this I would simply say something to the effect of, "I have an offer I need to either accept or decline by X. I would prefer to work with your client but with out an offer from them I am going to accept the other offer."
Recruiters will generally not get into a bidding war against each other. First it is not good business because in the end most of the time they would be bargaining against a ghost, second it cuts into their bottom line. You can sometimes leverage a single bump from an employer with a I will come to you for X. But I would not share that you have another offer. If they are willing to come up they will. If not then it will not really matter.
If you do share that there is an offer if you are working with a less scrupulous company they could potentially undermine that offer. Until you have accepted and the contract is signed you are not employed. If you have not accepted then they can rescind the offer. You could end up with no offers and two companies that no longer want to do business with you. You are far better served by keeping the existence of offers to yourself.