How are these compounds strong bases, when they are sparingly soluble in water?
The typical definition of strong base that you will see at an introductory level is that there is complete dissociation to hydroxide ions, for example that NaOH completely dissociates into Na+ and OH-.  

This is a terrible definition because reactions never go to completion.  The definition contradicts the concept of chemical equilibrium.  

In reality:

$\frac{[Na+(aq)][OH-(aq)]}{[NaOH(aq)]} = K_B = ~4M$ (from table discussed below)

So even dissolved aqueous NaOH is not all dissociated.  It can exist as ion pairs or molecules in solution without being dissociated.

$K_B$ is quantification of how strong or weak the base is.  Strong base could be defined as $K_B$ exceeding a certain value. Let us say $K_B&gt;1$ and therefore $pK_B&lt;0$

So the strength of the base only refers to the equilibrium between the undissoicated dissolved species and the dissociated ions.  It is independent of how much or how little solid dissolves. Some hydroxides, like NaOH and KOH have high solubilities and some have low solubilities, but $K_B$ is a different equilibrium.  

For quantitative data on the $K_B$ values of hydroxides, see:
DISSOCIATION CONSTANTS OF INORGANIC ACIDS AND BASES IN AQUEOUS SOLUTION
Look up the corresponding metal ion alphabetically, and a $pK_B$ value is given.  

However, keep in mind that for each of $\ce{Mg(OH)2}$, $\ce{Ca(OH)2}$, $\ce{Sr(OH)2}$ there will be two $K_B$ values, since there are two hydroxides that can dissociate.  

The $pK_B$ values for the second hydroxide dissociating are Mg 2.6, Ca 1.4, Sr 0.8, Ba 0.6 becoming stronger bases as you go down the column in the periodic table, increasing radius stablizing the charge of the ion.  The first $pK_B$ would be even lower, but I don't see any quantitative values.  It would be so low that it would be difficult to measure.  So, yes these are strong bases.

As far as how to tell, other than experimentally, how strong a base of a hydroxide is, there is a good lecture The Acid-Base Character of Oxides and Hydroxides in Aqueous Solution by Pilkington.  As you go from left to right across the periodic table (excluding the noble gases), the hydroxides progress from being strong bases to being acids.