What is a complete reaction?
I suspect that you would have both these molecular equations:

\begin{align}
\ce{2KCl(aq) + Na2SO4(aq) &amp;-&gt; K2SO4(aq) + 2NaCl(aq)}\\
\ce{Ca(NO3)2(aq) + Na2SO4(aq) &amp;-&gt; 2NaNO3(aq) + CaSO4(aq)}
\end{align}

To answer your question with "What is a complete reaction?":


  A complete reaction is one where all of at least one of the available reactants is used up and converted into products. 


So for example $\ce{KCl}$ and $\ce{Na2SO4}$: In this equation you will need $\pu{2 M}$ $\ce{KCl}$ and $\pu{1 M}$ $\ce{Na2SO4}$ in order for all the reactants to react. However, you only have $\pu{1 M}$ $\ce{KCl}$ (so $\pu{1 M}$ chloride ions). This means that, as you mentioned, you will need the double concentration for the $\ce{KCl}$ solution to achieve a complete reaction.

So in this case the only way for all $\ce{Na2SO4}$ molecules to react is to add the double amount of $\ce{KCl}$ solution. Then this is a complete reaction : at least one reactant is used up in the reaction.

But say we have a situation in which we have a $\pu{2 M}$ $\ce{KCl}$ solution, but a $\pu{2 M}$ $\ce{Na2SO4}$ solution, then all the $\ce{KCl}$ would react, but still \pu{1 M} of $\ce{Na2SO4}$ will not have reacted.If in your lab assignment it is the meaning that you react all $\ce{Na2SO4}$, then in this case you don't have a complete reaction. Only when a two times larger volume of the $\ce{KCl}$ is added (to 2M $\ce{Na2SO4}$), then you will once again achieve a complete reaction. Furthermore you would just get a solution of solvated ions.

In the case of $\ce{Ca(NO3)2}$ and $\ce{Na2SO4}$: In this equation you will need $\pu{1 M}$ $\ce{Ca(NO3)2}$ and $\pu{1 M}$ $\ce{Na2SO4}$ in order for all the reactants to react. You have $\pu{10 ml}$ of both compounds, each $\pu{1 M}$ in concentration: so in this case there is no need to use a larger volume of aqueous $\ce{Ca(NO3)2}$.