This question is addressed in ACI 301-99, "Specifications for Structural Concrete." We asked two members of the ACI committee that developed this document to comment on your question. Tim Moore is past chair of the committee, and committee member Dick Gaynor is past technical director of the National Ready Mixed Concrete Association.
Moore says concrete should normally be tested once, and the critical place to sample and test fresh concrete is at the placement point. The owner's testing laboratory is required to test fresh concrete either per ACI 301 Section 188.8.131.52.f (the default requirement-at the point of delivery) or per Section 184.108.40.206 (the optional requirement-at the point of placement if the engineer so requires for placement techniques such as pumping). If the point of delivery is different from the point of placement as in the case of pumped concrete, the testing laboratory should be testing concrete only at the point of placement. However, if the contractor is trying to establish an acceptable correlation and requires concrete to be tested at both locations, this would be the contractor's responsibility. If the owner wants to periodically check this correlation (previously established by the contractor) and test in both locations, it would be the owner's responsibility.
Gaynor says if the question is who should pay for testing at both places, he would hope that the parties--contractor, specifier, and concrete producer-wouldn't have to test at both places all the time. If that was necessary, however, the cost of the testing at discharge should be settled between the contractor and producer.
If the job specification requires testing at the point of placement, those tests will be the basis for the owner's acceptance or rejection. The specifier can't reject at both places without the actual or implied agreement of the contractor.
What happens with the producer depends on the purchase order with the contractor and whether the producer has done his homework. If the producer agreed to furnish concrete in compliance with the job specifications, he will likely be responsible. If the producer agreed to furnish concrete in compliance with ASTM C 94, "Standard Specification for Ready-Mixed Concrete," he should carefully consider the scope section of C 94, which says that the purchaser's specification governs. It also says that C 94 doesn't cover "placement, consolidation curing, or protection of the concrete after delivery to the purchaser." If the job specifications don't require testing at the point of placement and those job specifications include C 94 or ACI 301-99, tests at discharge govern. If the job specifications include ACI 301-99, testing at the point of placement may govern irrespective of what C 94 requires.
For more views from Moore and Gaynor on comparative testing, see "Testing Fresh Concrete: At Point of Placement or End of Chute?" from the April 2000 issue of Concrete Construction.