Introduction
It appears to me that this might be a bug. I certainly cannot think of any other explanation, but I would be glad if there is one.
The issue is with ParametricPlot, but I cannot explain why there should be a problem. Each Locator is basically an EventHandler that, when activated, tracks changes in position of the mouse and updates the coordinates of the Locator accordingly. What seems to happen is that a mousedown in one Locator activates all Locators when the graphics includes a ParametricPlot that depends on the Locator points. This somehow causes all dynamic locators to be updated whenever one dynamic locator is dragged.
It does not appear to be a localization issue or a kernel vs. front end variable issue -- that is, putting variables in Module or DynamicModule did not help.
As shown in other answers, the issue does not occur with LocatorPane, which creates a single event handler for the locators.
There is another issue in the OP's code. Since p is not localized and is set (several times) inside a Dynamic, there is continual updating.
One fix
Here is a fix of the OP's code that preserves the behavior of Locators, in case that is preferred over the behavior of LocatorPane.
Dynamic[With[{param = BezierPoint[t, ControlPoints]},
Show[
(* replace ParametricPlot *)
Graphics[Line[Table[param, {t, 0, 1, 1/100}]]],
(* localize p *)
Graphics[Table[With[{p = param}, {Text[t, p], Circle[p, 5]}], {t, 0, 1, 0.1}]],
Graphics[{
Line[ControlPoints[[1 ;; 2]]], Line[ControlPoints[[3 ;; 4]]],
Blue,
Locator[Dynamic[ControlPoints[[1]]]],
Locator[Dynamic[ControlPoints[[2]]]],
Locator[(*Dynamic@*)ControlPoints[[3]]],
Locator[(*Dynamic@*)ControlPoints[[4]]]}],
Axes -> None, AspectRatio -> Automatic,
PlotRange -> {{0, 400}, {0, 400}}]
]]
Another example and some evidence
Here is further evidence that the issue lies with ParametricPlot. This example from the Locator reference page works fine:
DynamicModule[{v1 = {2, 0}, v2 = {-1, 1}},
Dynamic @ Graphics[{Circle[], Red,
GeometricTransformation[Circle[], Transpose[{v1, v2}]], Green,
Line[{{0, 0}, v1}], Line[{{0, 0}, v2}], Locator[Dynamic[v1]],
Locator[Dynamic[v2]]}, PlotRange -> 3]]
But if we replace the circles with a ParametricPlot, we get an example similar to the OP's, and the locators are linked as in the OP's.
DynamicModule[{v1 = {2, 0}, v2 = {-1, 1}},
Dynamic @ Show[
ParametricPlot[{{Cos[t], Sin[t]}, v1 Cos[t] + v2 Sin[t]}, {t, 0, 2 Pi}],
Graphics[{(*Circle[],Red,GeometricTransformation[Circle[],
Transpose[{v1,v2}]],*)
Green, Line[{{0, 0}, v1}], Line[{{0, 0}, v2}],
Locator[Dynamic[v1]], Locator[Dynamic[v2]]}],
Axes -> False, PlotRange -> 3
]]
The locators are still linked if we move the symbols outside ParametricPlot and insert an evaluated parametrization:
DynamicModule[{v1 = {2, 0}, v2 = {-1, 1}},
Dynamic @ With[{param = v1 Cos[t] + v2 Sin[t]},
Show[
ParametricPlot[{{Cos[t], Sin[t]}, param}, {t, 0, 2 Pi}],
Graphics[{
Green, Line[{{0, 0}, v1}], Line[{{0, 0}, v2}],
Locator[Dynamic[v1]], Locator[Dynamic[v2]]}],
Axes -> False, PlotRange -> 3
]]]
Even if extract the Lines from the ParametricPlot, the locators are still linked.
DynamicModule[{v1 = {2, 0}, v2 = {-1, 1}},
Dynamic @ With[{plot =
ParametricPlot[{{Cos[t], Sin[t]}, v1 Cos[t] + v2 Sin[t]}, {t, 0, 2 Pi}]},
Graphics[{
Cases[plot, _Line, Infinity],
Green, Line[{{0, 0}, v1}], Line[{{0, 0}, v2}],
Locator[Dynamic[v1]], Locator[Dynamic[v2]]},
PlotRange -> 3
]]]
It happens with Plot, too.
However, the locators work independently if I define the function,
SetAttributes[myParamPlot, HoldAll];
myParamPlot[param_, dom_] := Graphics[Line[Table[param, dom]]]
and replace ParametricPlot[..] with myParamPlot[param, {t, 0, 1, 1/100}] in the OP's example or with two calls to myParamPlot, one for the circle and one for the transformed circle, in the modified doc. ctr. example.
The OP's approach seems reasonable to me, if Locator is preferred to LocatorPane. If LocatorPane is acceptable, then the Manipulate/DynamicModule approaches of other answers do not have the unexpected pitfall that the OP encountered.
