The idea behind publishers' data sharing mandates is to make data open for reuse. In most cases, data sets accompanying publications are freely available for reuse by other researchers, of course, with proper citation and attribution. Generally speaking, you are not required to take the permission of the authors, unless specified in the paper or the journal. However, you should use complete data citations in the style that is appropriate to your field. This usually includes the name of the author, publication date, title, publisher, and digital object identifier (DOI).
It is recommended that you check the style guide that is followed in your field while citing a data set. Here is an example of data citation formats used in different style manuals:
APA: Cool, H. E. M., & Bell, M. (2011). Excavations at St Peter’s Church, Barton-upon-Humber [Data set]. doi:10.5284/1000389
Chicago: (Footnote) H. E. M. Cool and Mark Bell, Excavations at St Peter’s Church, Barton-upon-Humber (accessed May 1, 2011), doi:10.5284/1000389.
(Bibliography) Cool, H. E. M., and Mark Bell. Excavations at St Peter’s Church, Barton-upon-Humber (accessed May 1, 2011).doi:10.5284/1000389.
MLA: Cool, H. E. M., and Mark Bell. “Excavations at St Peter’s Church, Barton-upon-Humber.” Archaeology Data Service, 2001. Web. 1 May 2011.<http://dx.doi.org/10.5284/1000389>.
Oxford: Cool, H. E. M. and Bell, M. (2011), Excavations at St Peter’s Church, Barton-upon-Humber [dataset] (York: Archaeology Data Service), doi:10.5284/1000389
Similarly, if you are using a data set from a repository such as Dryad, you should follow the data citation format suggested by the repository. Here is an example of the data citation format suggested by Dryad:
Kingsolver JG, Hoekstra HE, Hoekstra JM, Berrigan D, Vignieri SN, Hill CE, Hoang A, Gibert P, Beerli P (2001) Data from: The strength of phenotypic selection in natural populations. Dryad Digital Repository.doi:10.5061/dryad.166