See C.R. 103.4 ("First, the starting player decides whether they will take a mulligan.") and C.R. 103.5 ("[T]he starting player may take ... actions" called for by such cards as Leyline of Sanctity "in any order. Then each other player in turn order may do the same."). In both cases, in general, "[t]he player chosen to take the first turn is the starting player" (C.R. 103.2).
EDIT (Jul. 9, 2019): The text cited for C.R. 103.4 has changed in the meantime.
1) Opponent is on the play. We each draw 7. Who is the first to declare if they are going to mulligan or not?
2) Opponent is on the play. We each have a Leyline of Sanctity in our opening hands. Who places theirs into play first?
Thank you!
EDIT (Jul. 9, 2019): The text cited for C.R. 103.4 has changed in the meantime.