Your question raises a number of related issues - the preparer of the initial offer (which can be either the Buyer or the Seller) has the first shot at determining who pays what.
First, a qualification - the search itself is usually less than $100; the title insurance premium is the bigger number and, since it's based on the selling price, it should be lower IF the Buyer's payment of that cost results in the Seller reducing the selling price.
To a certain degree, whoever pays will also have a bit more "control" of the transaction; this may be an important consideration in a distressed property situation.
There are many more aspects to your question than any one responder (including myself) has covered; the "final" answer is that the contract determines everything that transpires after it is agreed to by both parties.
The common sales agreements used in our area allow for a variety of options that include both the buyer and seller being responsible for this service. Thus, this is one of those issues that can be negotiated between the buyer and seller.
Tammy Hayes, Green Lion Realty, Port Charlotte, FL firstname.lastname@example.org
Lynn911 Dallas Realtor & Consultant, Loan Officer, Credit Repair Advisor
The Michael Group - Dallas Business Journal Top Ranked Realtors