Virtually all major open source projects require contributors to agree to a contributor licensing agreement that includes language asserting that their contributions are able to be legally contributed to the project.
For example, Apache's Individual Contributor
License Agreement says
- You represent that you are legally entitled to grant the above
license. If your employer(s) has rights to intellectual property
that you create that includes your Contributions, you represent
that you have received permission to make Contributions on behalf
of that employer, that your employer has waived such rights for
your Contributions to the Foundation, or that your employer has
executed a separate Corporate CLA with the Foundation.
- You represent that each of Your Contributions is Your original
creation (see section 7 for submissions on behalf of others). You
represent that Your Contribution submissions include complete
details of any third-party license or other restriction (including,
but not limited to, related patents and trademarks) of which you
are personally aware and which are associated with any part of Your
Google's Individual Contributor CLA has virtually identical language (only the corporate names are changed). Other major open source entities probably have very similar terms.
This is a defense mechanism for the project: if someone sues the project for copyright infringement, they can easily deflect legal responsibility to the specific responsible contributor(s). Of course, despite this protection from legal liability, any infringing parts will still need to be removed or rewritten, or a compatible license must be acquired from the actual copyright holder.
As for "proof", though, proving the negative that a creative work does not infringe any copyright is an intractable problem. Not only does it require comparisons against the entire set of all copyrighted works (many of which are not publicly available), but the one-to-one comparison to decide that one work is a derivative of another is itself intractable. This problem is not unique to open source; it is a general problem with copyright law. Any time anyone offers you a creative work, for free or at cost, under FLOSS or non-FLOSS terms, you have no way to reliably determine that the distribution of that work does not infringe copyright.
However, in some jurisdictions (at least the United States) there is a legal notion of innocent infringement which may reduce or (rarely) eliminate liability for an infringer who had no reasonable expectation that they were committing infringement: for example, by distributing a contribution in an open source project from a contributor who had formally asserted that their work was not infringing.
If by "proof" you mean something like, "Contributor prompts their employer to assert that the contributor has permission to make contributions," I don't expect this to produce substantially better results than making the contributor sign and submit a legal document that asserts the same, and that plan would require a great deal of extra effort. It would be trivial for a bad or lazy contributor to say they have no employer, so this plan does not afford perfect protection, and the level effort involved (requiring both a cooperating contributor and a cooperating employer, even when no broad IP agreement even exists between employer and employee) would scare off even more potential contributors than requiring a signed CLA already does. Furthermore, it would allow an employer to unfairly block an employee's contributions in jurisdictions where broad employment IP assignments are not enforceable: the project would be waiting for the employer's say-so in accepting the contribution, but the contributor's local contract/IP law would not require them to do so.
Perhaps you underestimate (or I overestimate) the responsibility of contributors in making arrangements with their employers. I have signed a similar agreement, but my company has a well-defined exemption process, which I have used every time I made open source contributions.
Also, many projects offer corporate CLAs (e.g., here is Google's) so that a company can authorize all its employees to contribute to particular open source projects. In the particular case of open source contributions done by the employees major corporations for major open source projects (e.g., consider the extensive list of the corporate Linux Foundation members) where a corporate CLA exists, this is a non-issue.