Grouping list elements according to a condition

Select[Length @ # > 1 &] @ SequenceSplit[data, p : {{_, 0} ..., {_, 1} ..} :> p]

 {{{"Call", 0}, {"Call 2", 0}, {"Response", 1}}, 
  {{"Call 3", 0}, {"Response", 1}, {"Response 2", 1}}}

SequenceSplit[data, {p1 : {_, 0} ..., p2 : {_, 1} ..} :> {p1} -> {p2}]

{{{"Call", 0}, {"Call 2", 0}} -> {{"Response", 1}},
 {{"Call 3", 0}} -> {{"Response", 1}, {"Response 2", 1}}}

SequenceSplit[data, {p1 : {_, 0} ..., p2 : {_, 1} ..} :> {p1}[[All, 1]] -> {p2}[[All, 1]]]

{{"Call", "Call 2"} -> {"Response"}, 
 {"Call 3"} -> {"Response", "Response 2"}}

Update: An alternative approach using Split and SplitBy:

Rule @@@ Map[SplitBy[#, Last][[All, All, 1]] &]@
  Select[Length @ # > 1 &] @ Split[data, {#[[2]], #2[[2]]} != {1, 0} &]

  {{"Call", "Call 2"} -> {"Response"}, 
   {"Call 3"} -> {"Response", "Response 2"}}

crl={{"Call",0},{"Call 2",0},{"Response",1},{"Call 3",0},{"Response",1},{"Response 2",1}};

SequenceCases[crl,{c:{_,0}..,r:{_,1}..}:>First/@{c}->First/@{r}]

{{Call,Call 2}->{Response}, {Call 3}->{Response,Response 2}}

crl1={{"Bad Response",1},{"Call",0},{"Call 2",0},{"Response",1},{"Call 3",0},{"Response",1},{"Response 2",1},{"Bad Call",0}};

SequenceCases[crl1,{c:{_,0}..,r:{_,1}..}:>First/@{c}->First/@{r}]

Same as above.