A zero-inflated model has two components. Part 1 models if an observation comes from the "zero component" and is one of the "extra zeros". Part 2 is the "counting distribution", which is the model for how often the event happens in the ordinary observations. These two components can each either the same or different covariates.
For example, consider a study that looks at how often Men vs Women eat Beef per week. Some people might be Vegans and never eat Beef. Other people might eat Beef according to some distribution, but on a certain week they might not have eaten beef just by chance. This gives are two models. The "zero component" model examines whether men or women are more likely to be vegan and thus an "extra zero". The "counting component" model examines if there is a difference in how often the non-vegans eat beef. If we examine the possible relations between sex and eating beef, we could have:
- Sex has no relation on either being vegan or how often non-vegans eat beef:
y ~ 1 or
y ~ 1|1
- Sex has a relation on being vegan, but not on eating beef for those who aren't vegan:
y ~ 1|sex
- Sex has a relation on how often a non-vegan eats beef, but sex does not have a relation on being vegan:
y ~ sex|1
- Sex has a relation on both being vegan and how often non-vegans eat beef:
y ~ sex or
y ~ sex|sex
The situation you're asking about is #3. If that setting is the truth, it means that both men and women are equally likely to be a vegan, and thus be one of the "extra zeros". Another way to say this is that it doesn't matter whether you're a man or a women, you're both equally likely to belong to the zero component. That means that if we have a subject with an observed value of 0, we have no extra information on whether they're a vegan or not. Everyone who has an observed value of 0 is equally likely to be a vegan.
Contrast that with setting 2 or 4. In those settings we know that sex has a relation with being vegan. In those settings if we have a subject with an observed value of 0, we could look at their sex and find out more information about if they're vegan or not. We would be able to find out that some of the observed 0s are more likely to be vegan than others.
As for which is appropriate for your research, hopefully you have some idea from previous studies whether a covariate is significant on either component. If not you'll have to fit the model and perform hypothesis tests on the parameters for the two components.