The answer really depends on the license terms. There are massive differences.
In GPL v3, for example, there is clause 10 which says "Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License." This clearly indicates that if you just distribute unmodified source code, then the recipient will have the license from the original licensor.
In the European Union Public License (EUPL-1.2) you will find the definition of 'Licensor' to state "‘The Licensor’: the natural or legal person that distributes or communicates the Work under the Licence." So here, by virtue of distributing source code under the license, you become the licensor.
There are other FOSS licenses, which are silent about this point, or give both options, such as the MIT license, which provides (among others) the rights to "...publish, distribute, sublicense...", my reading in the case of publish and distribute is that you do not become licensor just by distributing the unmodified code, however when you sublicense it it is obvious that you are.