Question about cell signaling pathways (RTK, Jak-Stat, SMAD, etc)
@Alexandria Jak/Stat are two families of proteins which mediate signals through phosphotyrosines.  

JAK is a tyrosine kinase which binds to cell receptors and STAT is dimerized by JAK action.  JAK specificity seems to be your question.  A specific JAK protein (e.g. JAK1 or JAK2..) may mediate for different receptors in different cells. 

There may be hetero- or homo-dimers of JAK, there may be more than one receptor in a cell using the same JAK enzyme, which may result in cross-talk between the two signals.  I think its a pathway dynamic that can vary quite a bit from one cell to another, depending on the receptor and cytokine environment the cell is seeing. JAK action can be modulated by any number of proteins which respond to other pathway signals or the cell state.  The diagram below from a review of heart muscle cell JAK action is not unusual nor is it complete.