general relativity - Does a spinning mass follow a straight trajectory in empty space? Or is it curved like a golf ball?

In principle yes, but in practice they would follow the same trajectory. At least if you want to be able to actually use them for something useful.

The reason a spinning ball curves is due to its interaction with the air. If the ball is spinning clockwise as seen from above, and it is moving toward 12 o'clock, this pushes the incoming air a bit to the left, say, toward 7 o'clock.

By conservation of momentum, the ball must then move a bit to the right. This is called the Magnus effect.

For a ball of radius $r$, spinning at a rate $s$ and moving with velocity $v$ through a gas of density $rho$, the force is of the order (ignoring the dependency on the roughness of the ball's surface)
$$
F sim frac{16pi^2}{3} r^3 s rho v.
$$

In interplanetary space, the density is roughly $10^{-23},mathrm{g},mathrm{cm}^{-3}$. The Voyager space probes have reached maximum velocities of $sim60,mathrm{km},mathrm{s}^{-1}$ wrt. the Sun. Approximating the space probe as a spherical cow with radius 2 m (the disk is smaller, but the arms are longer), the force amounts to $sim (10^{-8},mathrm{N})$ times $s$. With a mass of $msim700,mathrm{kg}$, to accelerate it up to even 1 picometer per second per second, you would need to spin it at roughly 100 revolutions per second. In which case it would be difficult to use it for anything.

General relativistic approach

After your comment I realize you're interested in knowing about the effect of a rotating object on (empty) space itself, i.e. the effect known as frame-dragging. In the vicinity of massive, rotating object, space rotates along with the object. The closer to the rotating object, the faster space is "dragged along". A point-like test particle close to the object will start orbiting the object. If the test particle is extended, it will feel a "torque", causing it to rotate in the opposite direction of the object.

This means that two space probes sent off with opposite directions will speed up each others rotation, although this effect is miniscule for objects that aren't black holes. You can view the experiment from the frame of reference of their center of mass, in which they are stationary except for their rotation. The frame-dragging would look something like this:

Due to the symmetry, they wouldn't start rotating around each other, and would neither decrease of increase their inter-distance. However, due to regular gravitational attraction ($F=Gm_1m_1/r^2$), they would start to attract each other, and would in fact collide after the free-fall time
$$
t_mathrm{ff} = frac{pi}{2} frac{d^{3/2}}{sqrt{2G(m_1 + m_2)}},
$$
where $d$ is the distance between them, and $m_1$ and $m_2$ are their masses. For two masses of $700,mathrm{kg}$, separated by a distance of 10 m, they will collide in 32 hours.