Modelling on one Population and Evaluating on another Population
If the users who you are getting client-side data from are from the same population of users who you would get server-side data from. If that is true, then you aren't really training on one population and applying to another. The main difference is that the client side data happened in the past (by necessity unless you are constantly refitting your model) and the server side data will come in the future.

Let's reformulate the question in terms of models rather than web clients and servers.

You are fitting a model on one dataset and applying it to another. That is the classic use of predictive modeling/machine learning. Models use features from the data to make estimates of some parameter or parameters. Once you have a fitted (and tested) model, all that you need is the same set of features to feed into the model to get your estimates.

Just make sure to model on a set of features (aka variables) that are available on the client-side and server-side. If that isn't possible, ask that question separately.