Why Not Send Dumps?There is a long history of companies failing to minimize the data and to protect it. Microsoft for years sent crash dumps over the internet in the clear (WER & Privacy conerns). This allowed the NSA to harvest them, and develop 0-days for issues that MS failed to fix. Google's Chrome would send a screencap of the entire Desktop along with it's crash dumps (link). It previously would only send the window, but now sends the entire screen. Though they provide a preview, there is no way to see exactly what information will be sent.
I do not relish in advising people to not submit crash dumps as this will impact developers ability to fix bugs. But as with all aspects of security, companies continue to demonstrate that they are not willing to do the work that is necessary to protect user's data and their privacy.
CommunicationYou need to communicate to your users how crash dumps are handled. Just saying, trust us, does not inspire confidence, as there are a large number of cases of data breaches where the company has said exactly that leading up to leaks. The policy is the first step to demonstrating that you have thought about user's concerns and decided how you will handle their personal and sensitive data.
The policy also helps shape how employees will treat the data too. By having the policy, it is a reiteration to the employees that user data isn't simply chaff, but that it needs to be protected and handled with care.
How to handle dumpsThe first step is to ensure that what is collected in the dump has minimum information needed to debug issues. Code paths (back traces) are likely to be safe. Data, such as arguments to functions, may include user data and needs to be carefully examined. There are many different types of data that can be released from embarrassing (what website was visited), to security breach (including cookies/tokens for web sites that may not be yours), to confidential intellectual property leaking (source code, designs, etc). Each of these may have different impact on the user, but should never happen.
Second, crash dumps need to be transmitted confidentially. This means either using TLS or encrypting the dumps with a tool like GPG before sending. This ensures that unauthorized parties are unable to view the contents. The NSA used the dumps to gather information for their operations, which if Microsoft had properly protected their user's data, this would not have happened.
Third, they need to be stored in a secure manner and able to be expunged. It should even be possible for the user to remove the crash dump if they discover that information was shared when it should not have been. The life time that a company keeps the dumps should be limited. If you haven't fixed a bug from five years ago, how do you know you can reproduce it, or that if you are able to reproduce it, that the code is still present in your current software? It the crash is a major issue, it is likely that you'll have more recent dumps that exhibit the same issue if it is a problem, so old dumps are just not as useful compared to the data that may be present.
As crash data needs to be deleted, almost any cloud service is immediately excluded unless other precautions are used, such as encryption. With the cloud, you have zero visibility into how the data is managed and how or when it is backed up. Cloud providers rarely tell you their retention policies on back ups, and other policies that may keep data around. Do they securely remove your VM's storage when they migrate it? Do they ensure that storage is deleted from all clones, shards, servers and backups when you delete it? If not, how long will that data stay around before it is finally expunged.
Fourth, access to dumps need to be controlled. Auditing is a good first step to know who is accessing the data, but additional measures like limiting who has access needs to be used. Not everyone on the team needs access to them. As they are classified, they can be assigned to teams or people that need access to the data in them. This helps make sure that an employee isn't trolling for nudes or other confidential information. It should also limit how easy data is copied out of the archive. How these controls are put in place will vary by company.
Edit: Case in point: I recently opened a support case with Apple. Apple provides a program to collect data to send to them to help trouble shoot the issue. The program collected 280 MB of data. When uploading the data, Apple informs the user that it is their responsibility to NOT submit any personal information that they don't want. There is no way most people are qualified to look at the data, and even redact it properly. I attempted to do so, and it took a very long time, and I'm not sure that I got everything. Expecting a normal computer user to be able to do this is insane.