This question recently popped up on the front page,
so I entered $S_3$ in UACalc, and calculated
the $2$-generated relatively free group
$F_{S_3}(x,y)$. UACalc informed me that the order of this
group is $972=2^2\cdot 3^5$. (It gave me the multiplication
table too, but it was too big to be of much use to me.)
I then worked out the internal structure of the group,
using UACalc to check my intermediate calculations.

Next, I followed the zentralblatt link given in
Keith Dennis's answer, to see if what I'd done could
be found in Kovacs' paper. That paper predicts
the order of $F_{S_3}(x,y)$ to be $6377292 = 2^2\cdot 3^{13}$,
not $972$.

I started snooping around with google, and found a copy
of Kovacs' paper on his website with a red marginal comment saying that
his original formula is wrong. The marginal comment explains
how the formula should be modified, but unfortunately
the margin was too small to contain the proof of the correction.
Luckily for me, the corrected formula gives $972$ for the order of this group.

So, if you are interested in this problem, don't use the formula
in Kovacs' paper, but get the copy from his website.

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Let me make a conjecture about the structure of $F_{S_3}(n)$.
I have verified this conjecture by hand + computer
for $n =1, 2$, and it is
consistent with Kovacs' corrected formula for all $n$.
Moreover, the conjecture is so simple,
that if someone pokes at it enough they will likely 
find a proof or disproof.

Let $\mathbb F_3$ be the $3$-element field and let
$R = \mathbb F_3[\oplus^n \mathbb Z_2]$ be the group ring
with coefficients in $\mathbb F_3$
of the rank $n$ free group, $\oplus^n \mathbb Z_2$,
in the variety ${\mathcal V}(\mathbb Z_2)$ of elementary
$2$-abelian groups.

Let $V=\left(\oplus^{n-1} R\right)\oplus \mathbb F_3$
be the $R$-module that is a direct sum of
$n-1$ copies of the regular representation of
$\oplus^n \mathbb Z_2$ over $\mathbb F_3$ along with one
extra copy of the trivial representation.
Additively $V$ is an elementary abelian $3$-group,
but as an $R$-module it comes equipped with an action
of $\oplus^n \mathbb Z_2$ by automorphisms.

The conjecture is that the resulting semidirect
product $V\rtimes (\oplus^n \mathbb Z_2)$
is $F_{S_3}(n)$.

You can see from the description that the $\mathbb F_3$-dimension
of $V$ is $(n-1)2^n+1$, and so the group has order
$2^n\cdot 3^{(n-1)2^n+1}$. This is the modified Kovacs formula
for the variety ${\mathcal V}(S_3)$.
If the group I describe is $n$-generated, and if the
modified Kovacs formula is correct, then the conjecture
is true. I also note that the group I construct has the correct
size for its center and its commutator subgroup.