It is important to note that in the age of genetic engineering, traditional plant breeding techniques have not been abandoned. Those breeding programs are still used to provide a good base genetic stock into which to insert GMO (genetically modified organism) genes. The advantage of GMO techniques is that they are both faster and cheaper, because it is considerably less haphazard. This is especially true when trying to insert the ability to produce a protein from a different species, such as the Bt in insect resistant corn.
Because GMO corn still uses pollination to produce grain, there is a concern that GMO traits will migrate into non-GMO fields. One example might be when a field of non-GMO corn is right next to a field of Roundup Ready or Bt corn. The non-GMO plants might end up producing grain that has picked up partial genes that code for GMO traits. This would effect the grain in the ear of the corn plant, not the plant itself. And because corn seed is not held back for planting the following year this would be a one-time issue if it occurred at all. See the paragraph here about hybridization.
I used to grow non-GMO corn and it was exceptionally rare that we'd have problems with that. Non-GMO corn is tested for GMO traits when it is delivered to the buyer -- and we had very few loads rejected in the years we grew that specialty corn. We stopped because the premium we were paid did not come close to making up for the yield difference, not because cross pollination was a problem.
Cross pollination of soybeans is even less of an issue. Their flowers don't move pollen very far and produce a lot less of it over a shorter period of time -- compared to corn.
So called organic farming is simply not done on any scale at all with corn and soybeans near me. The closest I can think of would be growing non-GMO corn or soybeans for a premium contract -- which isn't done much anymore since Europe lifted their restrictions on American GMO crops. I do have (and buy from) a local organic vegetable producer, but there is no danger at all of cross species pollination from the GMO crops planted around their (really big) garden. Oh and for the record, I buy from this local organic CSA because they are local, not because they are organic.
Monsanto has sued farmers for violations of the seed technology agreement that the farmer signed in order to buy soybean seed, in which the farmer promised not to retain seed to plant the following year. If Monsanto tests leaf samples from one of my fields and those samples are definitively proven through genetic tests to be a specific Monsanto soybean seed, I had better have seed invoices to account for that. Cross pollination would not cause the plant itself to suddenly pick up some Monsanto trait, let alone change itself into a completely new variety. I haven't looked into many of these cases, almost all are settled out of court with sealed records. But my understanding of the testing methods used to build a case is that cross pollination wouldn't explain the genetic testing results.