(Initial post November 24, 2016, edited November 27, 2016) This does not exist.
The proof that $X$ doesn't exist is a bit elaborate and makes use of ends of coset spaces. I will prove:
(a) Let $\Gamma$ be a finitely generated subgroup of the group of isometries of the plane. If $\Gamma$ is not virtually infinite cyclic, then every $\Gamma$-commensurated subset $X$ of the plane is $\Gamma$-transfixed (see terminology below).
(b) For $\Gamma$ virtually infinite cyclic (that is, with an infinite cyclic finite index subgroup), a subset of the plane as prescribed does not exist.
Terminology (borrowed from here): if a group $\Gamma$ acts on a set $W$ (here, the plane), a subset $X\subset W$ is $\Gamma$-commensurated if $X\Delta\gamma X$ is finite for all $\gamma\in\Gamma$, and $\Gamma$-transfixed if there exists a $\Gamma$-invariant subset $X'$ such that $X'\Delta X$ is finite (clearly this implies that it is $\Gamma$-commensurated and furthermore that the cardinal of $X\Delta\gamma X$ is bounded independently of $\Gamma$).
(The link with ends of coset spaces is that for a finitely generated group $\Gamma$ with subgroup $\Lambda$, the Schreier graph has $\ge 2$ ends if and only if $\Gamma/\Lambda$ has a non-$\Gamma$-transfixed $\Gamma$-commensurated subset. Here restricting to transitive actions would be inconvenient, so I'm using the above language.)
First (a) and (b) imply the non-existence of $X$ as required.
We start from the classical fact that in a Euclidean space, any isometry between any two subsets has an isometric extension to the whole plane (see appendix C.2 in the Bekka-Harpe-Valette book, or check directly). So the question can be reformulated as follows:
(*) Does there exist a subgroup $\Gamma$ of the group of isometries of the Euclidean plane, generated by two isometries $\alpha,\beta$, such that, for some subset $X$ of the plane and $x,y\in X$ with $x\neq y$, we have $\alpha(X)=X\smallsetminus\{x\}$ and $\beta(X)=X\smallsetminus\{y\}$?
Indeed, take (a), (b) for granted, and assume by contradiction that $\Gamma$ exists. Then both generators commensurate $X$ and hence $\Gamma$ commensurates $X$. Since the first of this isometries maps $X$ to $X\smallsetminus\{x\}$, its $n$-th power maps $X$ to the complement of $n$ points in $X$: in particular the cardinal of $X\Delta\gamma X$ is unbounded and $X$ cannot be transfixed. So we get a contradiction, unless $\Gamma$ is virtually infinite cyclic, which is ruled out by (b).
Finally I'll justify that such "paradoxical" decompositions can occur without any reference to free subgroups:
(b') In the isometry group of the 4-dimensional Euclidean space, there exists $\Gamma$ and $X$ as in (*), with $\alpha,\beta$ generating a free abelian group of rank 2.
Let us first prove (a). The proof of the easier (b) is below.
Let $T$ be the group of translations in $\Gamma$. We distinguish two cases (the main one being the third).
1) $T$ is cyclic (either trivial or infinite).
Assuming that $X$ is not transfixed, since $\Gamma$ is finitely generated, by (Prop. 4B2 here), there exists a $\Gamma$-orbit $\Omega$ such that $\Omega\cap X$ is non-transfixed. Fixing a point $\xi$ in $\Omega$, write $\Omega=\Gamma/\Lambda$; that $\Omega\cap X$ is not transfixed implies that the number $e(\Gamma,\Lambda)$ of ends of the coset space $\Gamma/\Lambda$ is $\ge 2$ (by definition $e(\Gamma,\Lambda)>1$ means that $\Gamma/\Lambda$ has a $\Gamma$-commensurated subset that is not transfixed, or equivalently that is neither finite nor cofinite).
Then $\Gamma$ is a polycyclic group. Houghton (1982) proved that for a polycyclic group $\Gamma$, a subgroup $\Lambda$ satisfies $e(\Gamma,\Lambda)>1$ iff $\Lambda$ has Hirsch length 1 less than $\Gamma$ and has normalizer of finite index in $\Gamma$. Hence here, let $P$ be this normalizer and $P_+$ the set of motions in $P$ (of index at most 2 in $P$). Let $Y$ be the set of points in the plane fixed by $\Lambda$. Then $\xi\in Y$ and $Y$ is $P$-invariant.
Then $Y$ is an affine subspace. If $Y=\{\xi\}$, then $\xi$ is fixed by $P_+$, which is thus a group of rotations. Since $\Gamma$ is infinite, so is $P_+$, and so is some finite index subgroup $Q$ of $P_+$ that is normal in $\Gamma$; then $\xi$ is the unique fixed point of $Q$, and hence is fixed by $\Gamma$, a contradiction since $\Omega=\{\xi\}$ was supposed to contained a non-transfixed subset. If $Y$ is the plane, then $P_+$ is trivial, so $\Gamma$ is finite, a contradiction.
If $Y$ is a line, then $P_+$ acts by translations parallel to this line. Since $\Gamma$ is not virtually cyclic, $P_+$ is a finitely generated abelian group of $\mathbf{Q}$-rank $\ge 2$ and thus is 1-ended; it acts freely on the plane and hence transfixes every subset it commensurates, again a contradiction.
(Edit: see below for a more direct approach to Case (1))
2) $T$ is not cyclic (hence either it contains a free abelian group of rank 2, or a rank 1 infinitely generated abelian group of rank 1). A result of Scott-Sonneborn (Coll. Math., 1963), later rediscovered by Oxley (Math Z, 1972) implies that $T$ is 1-ended, in the sense that every subset of $T$ commensurated by $T$-translations is finite or cofinite [*]. Hence the intersection of $X$ with every $T$-orbit is either finite or cofinite. Actually this intersection is empty or the whole orbit for all but finitely many orbits [**]. Hence we can find a subset $X'$ with $X'\Delta X$ finite, that is $T$-invariant.
Since $T$ is normalized by $\Gamma$, $\Gamma$ permutes the $T$-cosets. Hence $X'\Delta\gamma X'$, for $\gamma\in\Gamma$, is a union of $T$-cosets, and thus either infinite or empty. Since $X'$ is commensurated by the $\Gamma$-action, it is thus empty and $X'$ is $\Gamma$-invariant, so $X$ is $\Gamma$-transfixed.
[*] From Scott-Sonneborn's paper, use Theorem 5, or from Oxley's paper, use Proposition 3.3 in rank $\ge 2$ and Proposition 3.11 in the infinitely generated rank 1 case.
[**] Suppose the contrary. Consider a countable infinite disjoint union $T\times I$ of copies of $T$ and for all $i$ a subset $F_i$ of $T$ (nonempty finite or a proper cofinite subset) such that the disjoint union $\bigcup F_i\times\{i\}$ is commensurated by $T$. Passing to complements, we can suppose that all $F_i$ are nonempty finite. Fix an infinite cyclic subgroup $Z$ of $T$. We can find elements $t_i\in T$ such that the $F_i+t_i\subset T$ are pairwise disjoint, and disjoint of $Z$. Then $\bigcup F_i+t_i$ is commensurated, infinite, disjoint from $Z$ and hence with infinite complement. This contradicts that $T$ is 1-ended.
Edit: here's a more direct approach to Case (1) in the setting of the question:
(1a) If $\Gamma$ has no nontrivial translation, then it's embeddable into $\mathrm{O}(2)$ hence virtually abelian. Then $\alpha^n$ is an element of infinite order and not a translation, hence a rotation of infinite order. Hence its fixed point is unique, and hence is fixed by some finite index subgroup of $\Gamma$, and hence by uniqueness by $\Gamma$. So the action fixes a point, say $O$. By removing entire $\Gamma$-orbits in $X$, we can suppose that $X$ is reduced to the (possibly equal) orbits of $x$ and $y$. One boils down to $\|x\|=\|y\|$ by possibly performing a similarity fixing $0$ to the orbit of $y$ (mapping this orbit to a subset disjoint from the orbit of $x$). Eventually $X$ lies in a cercle centered at $0$, and we can use the abelian case as in (2) unless $\Gamma$ is virtually cyclic.
(1b) the group of translations in $\Gamma$ is infinite cyclic: then it is normal, and hence centralizes a subgroup of index 2; since the centralizer of a nontrivial planar translation is reduced to the group of rotations, we deduce that $\Gamma$ has a subgroup of index 2 consisting of translations so $\Gamma$ is virtually cyclic.
Let us check (b). Then the "index character" $\tau:\Gamma\to\mathbf{Z}$ mapping $\gamma$ to $\#(X\smallsetminus\gamma X)-\#(\gamma X\smallsetminus X)$ is a homomorphism (this is a general fact). Here $\tau$ maps both $\alpha,\beta$ to 1; it is thus a surjective homomorphism and since $\Gamma$ is virtually cyclic, its kernel is finite, i.e. $\Gamma$ is finite-by-(infinite cyclic).
It is then easy to see that either $\Gamma$ fixes a unique point $\xi$ (which we can suppose to be 0), or fixes no point and preserves a line.
In the first case, $\Gamma$ acts freely on the complement of $\{0\}$. Being generated by two elements $\alpha,\beta$ of infinite order, it acts by rotations and hence is abelian. So we can get a contradiction exactly as in my comment for two translations (we can perform $y\mapsto\alpha^{-1}y\mapsto\beta^{-1}\alpha^{-1}y\mapsto\beta^{-1}y$).
In the second case, it preserves a line; since it is generated by two elements of infinite order, these act as nontrivial translations on this line and we deduce again that $\Gamma$ is abelian (action on this line, and hence on this coordinate is by translation, and action on the orthogonal coordinate is just by possible change of sign). We conclude again with the same trick.
Proof of (b'): for convenience, we use complex coordinates and work in $\mathbf{C}^2$. Fix an element $\zeta$ in the unit circle of infinite multiplicative order. Define the diagonal $\mathbf{C}$-linear isometries $\alpha(z,z')=(\zeta z,z')$ and $\beta(z,z')=(z,\zeta z')$. Define
$$X=\{(\zeta^n,0):n\ge 0\}\cup\{(0,\zeta^n):n\ge 0\},\quad x=(1,0),\;y=(0,1).$$
It is immediate that they satisfy the required hypotheses of (*).
Edit (August 25, 2018)
This (the non-existence of a planar subset as in the OP's question) was originally proved by E.G. Straus, On a problem of W. Sierpinski on the congruence of sets, Fundam. Math. 44 (1957), p. 75-81; available without restriction (at this time) here (Biblioteka Nauki); see also the Math Review link. The proof seems completely by hand and takes 4 pages.
