One area where there are multiple projects to share personal data in an open way is in genetics.
Projects like Genomes Unzipped contain autosomal DNA test results for named individuals (public domain). There are many more of these genotypes at OpenSNP. This personal data is shared to enable research on variations in genes for medical research, and also for ancestry (genealogy). There are many other ancestry-specific projects where named individuals share specific parts of their genes to hope to match with distant relatives, and some make their data freely available.
Some other data collections like 1000 Genomes do anonymise data, but there are problems when some projects try to keep personal data anonymous and others with similar or the same data do not, as the data can sometimes be matched to identify people.
Like Open Paths, Open Street Map has a large collection of GPS tracks. Some people use their full names, and the tracks probably include where they live and/or work, so they can be identified, even if that is not the intention.