I've got nothing on shadowing (I suspect that's a very hard sell), but for asking questions, there are two (non-exclusive) approaches that I've seen and occasionally been part of (on both sides).
The first is to start with your own circle of friends. I'm not a radiologist, lawyer, chef, schoolteacher, psychologist, soldier, horticulturist, priest, or a bunch of other things, but I have friends and acquaintances who are. Because I have social connections with them already (some stronger, some weaker), I can ask them occasional questions for reasons ranging from writing projects to personal projects to curiosity. Naturally, I should be just as willing to answer their questions about my areas of specialization. (Even if it means fixing their computers sometimes.) You probably have friends beyond your immediate circle of people you spend lots of time with -- think about your classmates, coworkers, congregation, Warcraft guild, Stack Exchange community, Google+ circle (oh, oops), etc. You know more people than you might realize, and some of them would be happy to talk with you.
The second is more conventional networking, where you don't know somebody with the right expertise but you know somebody who does and introductions are made. Finding some path to the expert you want to talk with works better than cold-calling. A "middleman" who knows both you and the expert can present your request, make sure the expert thinks it's reasonable, confirm that the expert has time and willingness, and then make the introduction if all that goes well. The expert never has to say "no" to you directly, and often your contact (the "middleman") won't have even told you who it is. This indirection is social-professional lubricant so nobody has to feel awkward. In this kind of networking, you ask someone who you think is likely to have the right type of connection to help you identify someone and make an introduction. For example, I would ask my radiologist friend to help me find a friendly brain surgeon; I wouldn't pick out a likely brain surgeon and go looking for someone who can introduce me. What do I know about brain surgeons, after all? My radiologist friend probably works with some and can steer me toward the friendly one or the one with spare time or the one who happens to share my interest in renaissance music.
That's all about connecting with someone. The other part is quid pro quo.
If you ask a friend for help then you can trade some other favor in return or just draw on "friendship capital". When you're instead networking your way to an expert, that doesn't help -- you probably don't have anything the expert needs and you're probably not going to be part of the same social circle where natural give and take applies. Instead, do something for the expert in proportion to the time and attention you're requesting. A common offer, from what I understand, is a nice meal during which you will ask and get answers to your questions. If you think you need more than a dinner's worth of time, I recommend choosing your most important questions, covering those in a dinner, and then asking if the person would be willing to meet again. Do be aware that in some professions, the person could be billing paying clients by the hour for similar kinds of advice -- stay well on the "information" side of the line and well away from anything that smells like seeking professional advice.
You might find that the expert is interested in your project, or enjoyed chatting with you and might want to socialize more, or believes strongly in mentoring and giving back and so is naturally inclined to help people in your situation. If any of that happens, great -- but don't expect it, and if the person gives the impression of wanting to limit the interaction to dinner and maybe a couple rounds of followup email, be gracious in accepting that. Depending on the scale of what you need, you might need to consult more than one person over time.
Finally, use your time with the expert respectfully. Do some research on your own before you ask for the meeting. Doing some research will help you ask better questions (and avoid the 101 stuff). Sharing your initial research establishes a baseline and demonstrates your investment in the project. You might not understand everything that the Wikipedia article on brain surgery says, but try to get the basic concepts yourself before your dinner with the surgeon.
(Humility works well here; just because you read it on Wikipedia doesn't mean you correctly understood it or even that it was right. Don't assume you know stuff; demonstrate that you've tried and show what you've gotten on your own.)
Where applicable, ask for pointers to information instead of information dumps -- two minutes spent on "where could I find an accessible description of X?" and an answer leaves more time for questions that can't be answered that way.
Understand your problem well enough to avoid asking for lots of irrelevant information. For example, do you need to know the details of how a certain brain surgery is conducted, or does your plot really revolve more around risks, time, and after-care? Know why you're asking the questions you're asking.