Step 1: Spot the shared group.
People who own both a cat and a dog are the same individuals whether counted from the dog side or the cat side, so we can equate the two counts.
Step 2: Count both-owners from the dog side.
$20\%$ of the $1001$ dog owners also own a cat: $0.20 \times 1001 = 200.2$.
Step 3: Express both-owners from the cat side.
Let the number of cat owners be $x$. Then $70\%$ of them also own a dog, which is $0.70x$.
Step 4: Set the two expressions equal.
\[ 0.70x = 200.2 \]
Step 5: Solve for $x$.
\[ x = \frac{200.2}{0.70} = 286 \]
Step 6: Verify.
$70\%$ of $286$ is $200.2$, matching the dog-side count, so the number of cat owners is $286$.
\[ \boxed{286} \]