For example, 350.org is a 501(c)(3) non-profit organization (according to its footer, bottom right) but signing up for its e-mail list or otherwise supporting it means a person would also wind up getting e-mails from "350 Action" (with list management still hosted at 350.org) making direct statements including "Vote for Hillary Clinton" and trying to recruit folks into canvassing on behalf of a particular candidate.
A key part of 350.org's methodology is "Pressure governments into limiting emissions" including "Run local and national campaigns targeting leaders." 350.org staff include people with job titles like "Campaigner," "Campaign Coordinator/Manager/Director," "US National Field Political Organizer," etc.
The distinction between 350 and 350 Action is not entirely clear, especially with use of the same mailing list. Further, the Privacy Policy linked to in the footer at 350action.org is just a link to the privacy policy posted at 350.org. Under "When We Share the Information," the policy says that they're "not interested in selling your information" but that they do share it with "the 350.org Action Fund (350.org’s affiliate organization)."
350 Action describes itself as a 501(c)(4) non-profit. At least two of its three board members are also on the board of 350.org, and the third is on the advisory council.
According to the IRS in 2016,
For an organization to be tax-exempt under section 501(c)(3) it cannot “participate in, or intervene in (including the publishing or distributing of statements) any political campaign on behalf of (or in opposition to) any candidate for public office.”
This includes not engaging in any biased efforts to get out the vote nor contributing to political organizations including maintenance of a separate organization under section 527.
Elsewhere, the IRS states more directly:
"Under the Internal Revenue Code, all section 501(c)(3) organizations are absolutely prohibited from directly or indirectly participating in, or intervening in, any political campaign on behalf of (or in opposition to) any candidate for elective public office. The prohibition applies to all campaigns including campaigns at the federal, state and local level...Political campaign intervention includes any and all activities that favor or oppose one or more candidates for public office. The prohibition extends beyond candidate endorsements. Contributions to political campaign funds or public statements of position (verbal or written) made by or on behalf of an organization in favor of or in opposition to any candidate for public office clearly violate the prohibition on political campaign intervention. Distributing statements prepared by others that favor or oppose any candidate for public office will also violate the prohibition. Allowing a candidate to use an organization’s assets or facilities will also violate the prohibition if other candidates are not given an equivalent opportunity."
The 350.org privacy statement says "We will not share your information with any individual or organization who is not engaged in furthering the success of 350.org." 350 wants to "stop all new fossil fuel projects" and keep all fossil fuels in the ground. Therefore, in order to remain in legal compliance with their privacy policy, they cannot give their mailing list or other resources to e.g. the American Petroleum Institute ("API's mission is to influence public policy in support of a strong, viable U.S. oil and natural gas industry") or its subgroup Energy Citizens ("people around the nation who support including oil and natural gas in our national energy strategy.") 350 can apparently point to this privacy policy clause as a reason why not sharing its list beyond its own 501(c)(4) and closely allied groups does not violate the "equivalent opportunity" requirement for receipt or use of its valuable mailing list and database about individuals, without losing its 501(c)(3) status.
There are also conservative charitable organizations supporting Trump, churches that may have slightly different rules, charities bearing the names of each candidate, etc.; this question is not limited to 350 but 350 just served as a conveniently concrete and sufficiently transparent example to help illustrate the kind of practices being asked about in this question.
What is the minimum degree of separation between a 501(c)(3) and its affiliated 501(c)(4) organization? Is it sufficient that a difference exist on paper (e.g. different articles of incorporation, different domain name, different bank account)? Are there restrictions against staff/directorial overlaps or the 501(c)(3) contributing its database/mailing list, etc. to its 501(c)(4) (including those that may have been satisfied behind the scenes)?
SE site selection notes: I recognize this might also be on topic at Law.SE, but I believe it's also on-topic here and this site has more specific domain experts who might be more knowledgeable about these specific laws. There seems to be a rule that a question asked on a site with a more general domain (like Law) about a specific topic that another SE exists for (like Politics), it would be closed on the former site without migration to the latter.