A pure strategy profile consists of functions $e_A(\theta_A)$ and $e_B(\theta_B)$ by which the players choose their efforts depending on their valuations.
First, note that the effort $1$ is dominated by the effort $0$, and the effort $0$ is dominated by very small efforts unless the valuation is $0$. Thus, at non-zero valuations the efforts will be in the interior of $[0,1]$, so equilibrium implies that the derivatives of the expected payoffs with respect to the efforts vanish.
The expected payoff for $A$, given $\theta_A$, is
$$
\int_0^1e_B^{1/2}\mathrm d\theta_Be_A^{1/2}\theta_A-e_A^2\;.
$$
Setting the derivative with respect to $e_A$ to zero yields
$$
e_A=\left(\frac{\theta_A}4\int_0^1e_B^{1/2}\mathrm d\theta_B\right)^{\frac23}\;.
$$
Thus $e_A=\lambda\theta_A^{2/3}$, and by symmetry $e_B=\lambda\theta_B^{2/3}$, and we can determine $\lambda$ by substituting above:
$$
\lambda\theta_A^{2/3}=\left(\frac{\theta_A}4\int_0^1\lambda^{1/2}\theta_B^{1/3}\mathrm d\theta_B\right)^{\frac23}\;,
$$
with solution $\lambda=\frac3{16}$. Thus, the symmetric equilibrium strategy profile is $e_A(\theta_A)=\frac3{16}\theta_A^{2/3}$ and $e_B(\theta_B)=\frac3{16}\theta_B^{2/3}$.
The expected total payoff is
$$
\int_0^1\mathrm d\theta_A\int_0^1\mathrm d\theta_B\left(\frac3{16}\theta_A^{1/3}\theta_B^{1/3}(\theta_A+\theta_B)-\left(\frac3{16}\theta_A^{4/3}\right)^2-\left(\frac3{16}\theta_B^{4/3}\right)^2\right)=\frac{999}{9856}\approx0.101\;,
$$
whereas if the players were to cooperate to maximize the expected payoff, they would each make the effort $e=\left(\frac{\theta_A+\theta_B}4\right)^{\frac23}$, resulting in a expected total payoff of
$$
\int_0^1\mathrm d\theta_A\int_0^1\mathrm d\theta_B\left(\left(\frac{\theta_A+\theta_B}4\right)^{\frac23}(\theta_A+\theta_B)-2\left(\frac{\theta_A+\theta_B}4\right)^{\frac23}\right)=\frac{2916\cdot2^{1/3}-1899\cdot2^{2/3}}{6160}\approx0.107\;.
$$
Thus the players come surprisingly close to the cooperative result.
