Since the components of the given planar RBM are uncoupled, it suffices to consider a scalar RBM $X$ on say $[0,1]$.  Here is a sample path of this process on the interval $[0,10]$.

![sample path][1] 

I will approach local time via the occupation times formula 
$$
\int_0^t 1_{[0, \epsilon]}(X(s)) ds
$$
which gives the (random) amount of time the process spends in an $\epsilon$ neighborhood of zero during the interval $[0, t]$. This is a fairly complicated random variable, since it depends on the entire path of the solution process!  However, its expected value is analytically available.  Let
$$
u^{\epsilon}(t,x) = \mathbb{E}_x \int_0^t 1_{[0, \epsilon]}(X(s)) ds
$$ Note that $u^{\epsilon}(t,x)$ satisfies an inhomogeneous initial boundary value problem:
$$
\partial_t u^{\epsilon}(t,x) = 
\frac{1}{2} \partial_{xx} u^{\epsilon}(t,x) + 1_{[0, \epsilon]}(x)  \quad \forall x \in [0,1] \;, \forall t \ge 0 \;,
$$
with initial data $u(0,x)=0$ and pure Neumann boundary conditions $\partial_x u(t,0) = \partial_x u(t,0) = 0$.  By expanding the solution and the inhomogeneity in terms of the orthonormal eigenfunctions $\{ e_j(x) \}$ of the second derivative operator on $[0,1]$ with pure Neumann boundary conditions at $0$ and $1$, one obtains the following explicit solution:
$$
u^{\epsilon}(t,x) = \epsilon t + 2^{3/2} \sum_{j \ge 2}  \left( 1- \exp\left( \frac{t}{2} (j-1)^2 \pi^2  \right) \right) \frac{\sin( (j-1) \pi \epsilon)}{(j-1)^3 \pi^3} e_{j}(x)
$$   
The figure below plots the behavior of $E_x L_t := lim_{\epsilon \downarrow 0} \frac{u^{\epsilon}(t,x)}{\epsilon}$ as a function of the initial condition $X(0)=x$ with $t=1$.  As expected, this quantity decreases with distance from zero.

![local time][2]


  [1]: https://i.sstatic.net/s06YX.jpg
  [2]: https://i.sstatic.net/T8jnC.jpg