The first thing to realize here is that even in academia, not all studies are funded directly by the public sector. Private corporations frequently fund studies that are of interest to them. Results of such studies may bring up conflict of interest questions down the road, but most importantly the private options is a viable alternative.
(from here on I'm just going to assume US)
Political campaigns can also go both ways. If a campaign does not want to raise funds from private donors they can opt to take public funds from the FEC. These public funds are fed by the optional $3 tax user4012 mentioned in his comment. In reality though, most candidates will not choose this because it ends up being less money overall
[Public funding] also placed limits on campaign spending by Presidential nominees who receive public money and a ban on all private contributions to them.
Private funds, unlike public funds, are virtually limitless and carry much fewer restrictions. The bottom line is that its easier to raise more money with private funds than with public funds.
There is also a legal issue with banning private funds from political elections. The defining case for this is Citizens United V FEC. Simply put, Citizens United stated that money, especially in the form of political contributions, is a form of speech. Since the first amendment prohibits restrictions on speech by the government, the government is not allowed to restrict political contributions. So by banning private funding you would be banning a form of speech... a big no no in the US.