What is the best way for synthetic data generation while maintaining privacy?
Whichever method you choose is fine but assuming that you wish to mitigate inference attacks something like differential privacy is required for either approach.
Formally speaking, differential privacy provides some of the strongest guarantees against reverse engineering. Specifically, it promises that any attacker, regardless of attack methodology or available computing power, will be unable to conclude with certainty whether or not any individual has contributed data to a dataset. This is because the results of differentially-private methods are ambiguous up to the addition or removal of the input contributions of any individual. In essence, every individual gets deniability about their participation (or non-participation) in the input.
The problem with synthetic data is that it is generated from a model that is fit to real data. This means that model parameters are aggregate functions of the real data. This is problematic as it is often possible to make inferences from aggregates of data or estimates thereof (this is the motivation for differential privacy in the first place), and parameters of the generative model can be often be estimated from the synthetic data. I am happy to give an example of such an attack if there is interest. Further, it should be noted that this reasoning also implies that white-box exchange of the model is at least as risky and comes with additional concerns like has this network memorized training data. A straight-forward mitigation for this is to apply differential privacy in building the generative model for synthetic data.
In regards to privacy budgets, one can interpret the budget (often referred to as $\varepsilon$ and also the privacy-loss) as the amount of information (say in bits) which can be inferred about any individual by an adversary who has access to differentially-private results. Perhaps surprisingly, it can and ideally should be much less than 1. If there are future updates which reference the same individuals then one has to worry about how much individual information can be inferred from the aggregate collection of releases. There is a straight-forward composition theorem (See e.g. Sect 3.5) that follows directly from the definition of differential privacy. It states that aggregate privacy-loss is at most the sum of the individual privacy-losses of the constituent releases. In other words it's additive in the worst case. It may also be helpful to know that when the inputs are disjoint it goes like the max.