Making data sets available for re-use by other researchers is an established scientific methodology that can reduce costs for research. However, there is not yet consensus about how to best publish or make available data sets, what counts as "re-use," and not all publications include adequate citations to data sets (so undercounting of use is frequent).
If you are a researcher who has made your data available for re-use, define for yourself what you consider "impact": a valuable and substantive re-use of your data. Try to find this information, and include this as a surrogate measure of your individual researcher impact.
Data sets can be deposited by researchers in a variety of types of repositories: US-government-run repositories, free subject-specialized repositories set up by professional organizations, free, multidisciplinary repositories set up by nonprofits, University-run repositories, or a personal website.
Some of these sites will include frequency counts for how often a data set has been viewed or downloaded. Additionally, one source tracks use of repository data in articles.
This database harvests records of datasets from many repositories, and tracks the number of times these data sets have been cited in journals in the Web of Science database.
Impact of data papers
Data can be published in journals: "data papers" or "data description papers" describe datasets in depth in order to encourage re-use. Other journals will require or allow a dataset as an appendix when publishing a paper, so any citations to the data will refer to the article citation. For general tools for tracking the impact of articles, review the Article Impact tab. There are also some specialized databases that track citations to data published in journals and the re-use of these datasets.
Search Scopus for the name of the dataset of interest, then select Secondary Documents on the search pane. Look for your dataset and the number of times it has been cited by articles included in Scopus.