My guess is that it would depend upon the contract, the pre-caster and the capabilities of that pre-caster. I worked as a design engineer for a precast manufacturer in Ft. Worth for about 2 years. We did mostly prestressed/precast parking garages but site requirements sometimes forced us to design other structures (such as retaining or barrier walls). We had a design office that was composed of about 7 structural engineers and a drafting department with about 20 draftsmen as well as a construction yard with all of the formwork, double tee beds, concrete batch plant, etc. required to produce the product. We would design everything (shearwalls, stair towers, L beams, T beams, double tees, columns, etc. including lifting inserts, additional reinforcement for those inserts, etc.) from the foundation up as per the code in force for the location of the project which meant both lateral and gravity loadings as well as the connection designs between the precast elements, the shearwalls and columns to the foundation, etc. We needed to be informed by ether the Architect or EOR of any special loading conditions (I once had to design parts of the top level of a parking garage to support planters that would carry three feet of soil so they could plant small "growies"). We would provide (if requested) gravity loads, reactions, overturning forces, etc. to the EOR for his design of the foundation. In addition to the construction tickets for our own people in the construction yard, we provided stamped design and erection drawings as well as stamped calculations for the review and permit process. The line of separation between us and the EOR was the top of the foundation although, as I said, we did provide the design and detailing for the foundation connections of the precast elements. I'm not sure in the end who was ultimately responsible for the structure since we did provide a stamp for the design of the structure itself - that was above my paygrade at the time. I am sure that in the end, if the lawyers got involved, that it would make no difference - everyone involved would get pulled in to a lawsuit.
If you are talking designing a pre-cast structure as a tilt-up type structure to be built on site, then it is you who designs eveything except that the contractor should be responsible to take care of any modifications to the pieces that may be required for lifting and erection (inserts, stiffeners, etc.).