Leakage between requests is only one of the problems with globals. There's also poor testability, inability to re-use functions that reference the request internally, inability to statically analyze dependencies (less of an issue in Python, since the lack of types makes it hard to analyze them anyway), additional context that the developer must keep in his head, and inability to cross-reference parameters & call-sites and identify where a given variable is coming from in code browsers or debuggers.
Singletons are no better - if you dress it up as RequestManager.get_current_request(), it's just as bad. If you don't understand why something is a bad idea, complying with the letter of the law without complying with the spirit doesn't really get you much.
I think this is the way in which my programming style has changed the most over the last 10 years. I used to want my programs to do everything and have access to everything. So I'd write PHP scripts that would do "SELECT * FROM table INNER JOIN table2 WHERE ..." and pass around the whole result set to every function and every template in the app in case they wanted to access some data in it. That way, I only had to change the DB schema and the template when requirements changed. (Anyone who's thinking of doing this, you're introducing massive data leakage and security issues into your app.)
Now I want my programs to do as little as possible and have access to as little as possible. Usually I end up pulling out the values I want inside the view function and calling any helpers using only those values, so I don't pass the request around anyway. The actual work of the program happens on strings, or ints, or structs, or other values specific to the domain.
I think what changed, for me, were a few things. One is that I got better at visualizing the operation of a large program as a whole, so instead of thinking "Hmm, I need a request here, how can I get access to one", I started thinking in terms of "This is how data enters the system, this is what we need for each computation it performs, and this is how we combine those results to render a page - how can I change it so the system still hangs together but does what we want now?" Another was that I kept running into instances where I could almost re-use a function, but was blocked because of hidden dependencies that it didn't really need. A third was testability - I learned how crucially important it is to be able to test parts of the system in isolation, and that's really hard when you have to setup the whole system context to run any function. A fourth was that I worked a bunch with both systems and found that adding a parameter to each function along a call path is easy (but tedious - if only there were a refactoring tool to do this for us), while disentangling a tightly-coupled function that depends upon a lot of implicit state is quite challenging.