The "common term" would be in standard form, but the two input multivariate polynomials needn't be, e.g $x(1+y)+y$ and $y(x+a+b)$ have one common term, $xy$.
A brute force solution would be to expand both polynomials to standard form, then check for common terms. e.g. $x+xy+y$, $xy+ay+ab$.
Is there a better way?
EDIT if it makes it easier, we can loosen "same term" to allow different coefficients, so that e.g. $2xy$ and $3xy$ count as the "same term". (there must be a more mathy way to say that)... so, perhaps it becomes like a single component in Fourier analysis?
Maybe, a specific term could be identified by differentiating its factors by their power number of times to eliminate it (e.g. $x^2y^3$, differentiate by $x$ twice and by $y$ three times). Though, seems inefficient, and unclear how to extend that to checking all terms apart from brute force.
BTW "no common terms" is the opposite of "all terms in common" (in a sense), i.e checking for identical polynomials. This does have an efficient algorithm, using the Schwartz-Zippel lemma (see Is there an efficient algorithm for expression equivalence?).