How to generate a random point on an elliptic curve without knowing it's corresponding scalar private key
Pick a random $x$ value. Calculate $y^2 = x^3+ax+b \bmod p$. Then try to form $y$ by taking the square root $\bmod p$. If the square root fails then no $(x,y)$ pair exists on the curve. If the square root works, flip a coin; if tails form $y = p-y \bmod p$.
This is how public key compression works. Only the low bit of y is saved. Form $y^2$, take the square root (which had better work). If the low bit of $y$ is wrong then form $y = p-y$.