Efficient Identity Based Parameter Selection for Elliptic Curve Cryptosystems by Arjen K. Lenstra contains a proposal for a non-shared elliptic curve cryptosystem. Every party chooses its own field and then after some computations and checks (Table 1 in paper) chooses Weierstrass model. The main reason of this work is minimize of trust to the third parties (don't use pre-selected elliptic curves).
The same Weierstraß model used over two different finite fields gives rise to two different and, from a security point of view, independent groups. With the current algorithms, ability to solve discrete logarithms in one of those groups does not make it easier to compute discrete logarithms in the other group.
Question: How is possible that parties do not have to have the same group? I thought that having the same group is essential. How can I perform any operations correctly, if the chosen groups are not identical?