It's an excellent question with which I too am struggling. My current concern is only for intranet situations where we control both client and server. In these situations my feeling is that it's best to take the middle ground. I think it is important that another developer can look at the .aspx and understand the gist of the page.
For example, if a table of data needs to be displayed, then instead of sending it out as json and then templating it on the client, which would not show intent very well, send it as an ordinary ASP.NET datagrid html fragment. The datagrid sitting on an .aspx at least shows the intent to display tabular data. And it's probably easier to do it that way as well.
Now, suppose you want to display a really fancy datagrid, that would be difficult to do using just ASP.NET. I would still send it as a simple datagrid, because jQuery can slice and dice it a lot more easily on the client than ASP.NET could have back on the server. And probably this will save some bandwidth as well. (Oh, I ALWAYS turn off ALL ViewState!) I try to let ASP.NET do the semantics, and the easy stuff that it is good at, and let jQuery do the rest.
In practice things get more complicated and fuzzy. As a long time Model View Presenter guy, I am starting to think of the View and Presenter as living on the client. The .aspx page is just a template that shows the gist of the page. And the code-behind (.aspx.cs) is nothing more than a thin layer that delegates to the Model.
While I have been very happy with the results of this architecture, still I feel vaguely uncomfortable about it, mostly because I can't unit test the presenter with trusty old NUnit, and also because jQuery and JavaScript are so powerful that it's like playing with dynamite.
The usual architecture experts are silent, this is all too new. I'm 100% sure that many have already studied exactly this kind of situation as long as 30 years ago, and sooner or later this will all be discussed at length everywhere you look. But it's a very interesting situation we are in right now, I wish more people would put in their 2 cents. Where's Martin Fowler? Scott Bellware? Uncle Bob?
Mike