How can calcium be a kept as a local signal in the cytoplasm?
As @BryanKrause says in his comment, net flux of calcium would not ever be from inside the cell to outside the cell. There is a large gradient the other direction. Per Berg, the cytosolic concentration of calcium in a resting cell is on the order of 100nM, while extracellular concentration is in the mM range.

Signalling occurs when channels open up and allow $Ca^{2+}$ to diffuse down that very steep electrochemical gradient (from extracellular or topologically similar sources, e.g., the endoplasmic reticulum). 

At least part of your question seems to involve a concern that the signal would not be very long lived. Though your concern about it diffusing out of the cell isn't warranted, the question about the time dynamics of calcium signalling is interesting. Calcium signals are, in fact, often very brief (though in other cases they last much longer). This is actually a benefit, and is one reason these signals can have very localized, very brief effects. For localized membrane fusion events, e.g., exocytosis, the signal lasts on the order of microseconds. The control of the time and position of a calcium signal involves a large number of players, including sequestering proteins, various compartments and stores, and tight control of the balance between calcium on and calcium off signals. It is discussed in this nature review, and summarized in this simplified figure:



You might also be interested in looking at some videos of calcium signalling. There are a number on youtube. This is one example