Asking for data as a student - ease the pain
There are similar questions on this site for students requiring source code, such as What action to take when questions regarding a published paper are ignored by its author?. In your case you need data but as you have already seen yourself, sharing data is hardly a volunteer action.

Sharing data is a form of collaboration and collaboration needs equal (or almost equal parties). You benefit from taking the data but how does the other party benefit? You will probably say that the other side will increase his citations but how you guarantee that you will actually get something published if you are just a graduate student at his beginning? When your professor contacts them, two things are guaranteed: a) The confidentiality of the data b) The fair use of the data, including citation. When you contact them, none of these two conditions apply.

Second, many times graduate students just collect data without a clear understanding of what to do with it. It is easier to send a dozen emails to request all sorts of data than actually getting job done on the ones you have. You said it yourself: 


  ...tons of interesting data is now freely accessible over the web


So, why do you need extra data, when there are already many available datasets for you to use? Have you already published something with the freely available data? Have you even compiled a similar dataset? Would you actually share your dataset if you had one? If the answer to most of these questions is NO, then these are the things you must focus on and not waiting to get another dataset for the sake of it.

Also since collaboration requires equal partnership, you will see that these things get easier when you have established yourself, even as a junior researcher. If you have some good publications (using the freely available datasets or the ones that your professor got for you), then when you request additional data (on a polite email introducing yourself and leading to your homepage and google scholar profile), you will not get ignored, because the dataset holder will know two things: a) That you know your craft and his dataset will be used in a fair way (citation included) b) That you may collaborate in the future, so saying NO to you is not beneficial to him. Until then, I do not think there is anything much you can do than using your professor. But since this has already happened more than once (and there are many freely available datasets), I doubt you really need all these datasets you are collecting.