Is it possible to use a generative model to "share" private data?
Unfortunately I don't think a generative model could prevent from leaking private information from the original dataset.

Like any other kind of model, the generative model is based on the values obtained from the training data. The idea of using such a model in "generation mode" is indeed interesting since it would make it difficult to reverse-engineer the instances it generates back to the real individuals. It would be difficult sure, but not impossible: by re-connecting pieces of information together or exploiting rare (distinctive) cases, somebody could acquire at least partial personal information from the instances.

Additionally the design of the generative model itself introduces a huge bias in the data: the distribution of the instances would be modeled after this design, which might or might not accurately represent the real distribution. This issue significantly lowers the interest in exploiting the generated instances, since it's essentially artificial data.

For the record I think I've seen research about using distributed ML methods in order to overcome the privacy issue. As far as I understand the idea is to keep every individual/institution in control of their data, allowing specific automatized methods to read it in some kind of safe way.