There is undoubtedly the need for tremendous amounts of research into human physiology and health, but there's a very naive hubris that runs through all these grand analysis projects that promise to solve all human health woes. Our bodies are the most complex systems that we know of in the entire universe. The idea that we are anywhere close to understanding how they should optimally work is just completely daft.
Just look back at the promises made by the Human Genome Project in the early 90s. Yes, we've learned a lot since then, but mostly we've learned that the relationship between our bodies and our genes is far far more complicated than we imagined 25 years ago.
They should consider partnering with these guys: http://www.nature.com/news/medicine-gets-up-close-and-person....
Maybe the thing behind this google move is an understanding that it's possible, in some time feasible for google's projects, to commercialize such testing platforms, and they want in at the opportunities that will result ?
Heck, when I donate my blood I am already giving a lot of information to the blood center and they know my blood type and can tell if I have certain diseases of not. When I volunteer to do bone marrow match test, I don't know how the firm handles my data. I supposed they can protect my data.
I am not worried about Google knowing more about me or humans. I am more worried about how safe my records are with my clinics and hospitals. See 
Lastly, personal question: how do you get into Google X?
Now that I read what you wrote twice I might interpret your original sentence wrong. I thought you were implying the work they doing with Google is using public money ("I prefer research funded by public money [done by these researchers to be non-profit].")
You meant you prefer to have public institutions to carry out the research.
Well, in that case, it's only temporary. There are public and private AIDS research, but in the end, the public research is used to seek for profit. Say Google, IBM, Microsoft donates 10B to make this a non-profit research, the end product (the algorithms, etc) will be used by private companies anyway. So I don't see how public vs private really matter if the data and results are going to be publicly known by private company later on for profit.
But if you are going for "pure knowledge", sure, public research would be ideal :D
I was just googling data breach for hospital; not rare.
I say, wouldn't it be amazing if everyone's medical records were available anonymously to research doctors? And when someone accesses your medical record, a research doctor, they could see, you could see which doctor accessed it and why, and you could maybe learn about what conditions you have. I think if we just did that, we'd save 100,000 lives this year.
So I guess I'm just very worried that with Internet privacy, we're doing the same thing we're doing with medical records, is we're throwing out the baby with the bathwater, and we're not really thinking about the tremendous good that can come from people sharing information with the right people in the right ways.
Source: https://www.youtube.com/watch?v=mArrNRWQEso&t=839 at 14:00.
Much of this internet privacy backlash is aimed specifically at Google and Facebook (and of course the NSA, but let's assume that they don't have any problems getting to your medical data anyway).
But then even if they weren't anonymized, I think Google would be one of the company I would trust the most for that. They know what they're doing in terms of security and of all the companies that know a lot about me, they're the one who consistently delivered great value from the data they have. I'm thinking of things like search, Google Now and the maps interface for my location history. Plus I'm happy that the monetization of my data serves to fund great projects like this one. It didn't have to be so...
Health data is very far from this idealized anonimization-friendly case. It is some of the most specific data avalible about you, and it can easily be abused to disadvantage you in ways you can do nothing about (insurance, job prospects, social standing, even as far as getting your life threatened by people who disagree with things like sexuality and abortion)
Im sure Google appreciates your faith in them. But you should carefully consider the consequnces. Once the information is out there, there is no getting it back.
 Although Google's direction is arguably more stable than is usual for publicly traded companies, given the particular share structure designed to keep control in the hands of the founders.
 Mostly, being in generally good health and living in a country where basic health insurance is guaranteed and good enough, with premiums that don't depend on your health.
The only way you can properly anonymize such data is by reducing it to aggregates but then most of the utility is lost.
Taking the combination of google search data and medical data it would not at all be hard to tie some of the searches (and thus identified users) to medical records. Especially not if the diseases / symptoms are rare.
Researcher should be forced to follow ethical guidelines in order to prevent data mining that try to unearth facts written between the lines, like sexual orientation. Patients should also be asked if the journal can be shared with researchers, in similar fashion as asking patients if they will accept to have an intern in the room. Last, there need to be some liability if journals get leaked. It is very important to create economic incentives for functional security, which experts are paid to regularly verify and test.
This has of course nothing to do with keeping large databases about people and selling it as a service or product. Nothing at all about Google business model.
As for medical records, no institution hoarding them has any interest in sharing them. Patients are not consulted on this, so their privacy is only a tiny part of the reason. As far as patient privacy goes, “anonymising” them is not so easy either, as another commenter already explained.
EDIT: Seriously, if the yardstick is "but but but think of all the great things we could do, just like we have with the internet" I'm going to request proof that we've in fact accomplished that tremendous good in terms of advertising and whatnot.
If they can't even show that for their main line of business, any claims about medicine seem shaky, no?
Are we going to end up with an average that doesn't fit anyone?
Google has shown itself to be driven by metrics to almost the sole exclusion of anything else, metrics often dont tell you what you think they do.
The thrust of this approach (as mentioned in the article) is that they're searching for the magic of optimal biomolecular processes, not necessarily a universal regimen.
Knowing a little bit about how data moves thru the Labs &
clinics systems however, and the widespread tentacles of bad actors like the NSA, this could be too juicy a prize to ignore.
What is to prevent the Feds (for instance) from knocking on their doors with an NSL for the data based on some trumped up link?
"23andMe Is Terrifying, But Not for the Reasons the FDA Thinks", https://news.ycombinator.com/item?id=6811167
"23andme has suspended health-related genetic tests",
"A problem with much statistical analysis is ignoring the fact that humans, umm, react to things around them. (Social science jargon for this is reflexivity). I know this seems so simple, but it’s amazing how much predictive analytics don’t factor this in.",