When we check for pieces of research, for example through the Pubmed database, how can we check the source of funding? (And check for possible financial conflict of interest?)
Reputable journals might require authors to disclose funding sources, but as noted, this practice, reputable journals notwithstanding, is inconsistent. Beyond this, there are no standards for determining what is a reputable journal.
As to the comment that two citations do not prove the point, the literature is replete with studies demonstrating funding bias; a few examples: Pharmaceutical industry sponsorship and research outcome and quality, Lexchen, Bern et al; Avoiding biasing the conduct and reporting…, Hillman, Eisenberg, et al; Systematic review of empirical evidence of study publication bias, Dwan, Altman et al.
I wouldn't say that two studies is "proven," especially when neither of those is empirical: The first one is a literature review, and the second one is a theoretical article. Besides, both of them talk almost exclusively about the funding of medical/pharmaceutical research by pharmaceutical companies, and not about funding agencies as a whole.
That said, journals are supposed to be the gatekeepers of this information. When scientists submit a journal article, most journals these days ask them to disclose a conflict of interest: whether their funding or other personal interests is tied up with the research they're doing. Some journals publish this information publicly for readers to see, and others do not. Some journals require a funding statement - a little paragraph in the footnotes that states where the funding comes from
But this is done inconsistently; there's no standard from journal to journal, and not all journals report the information at all. The one "exception" is NIH funding - because of new NIH regulations, scientists have to disclose their NIH funding in the journal article and link their funding to the articles themselves, so theoretically the public can see where their tax dollars are going. In practice, the process of linking can take several weeks to several months (although this will probably speed up as it becomes more commonplace).
So the answer is - often times the only way to check IS what authors themselves report in the paper, short of contacting the author(s) and asking them yourself.