Dear unknown, here is a sketch of proof for your question.

Suppose there were a point $v\in V_1\cap V_2$ which was holomorphically non singular on $V$, i.e. holomorphically smooth. Then the germ of analytic space $V_v$ would have a decomposition $V_v=(V_1)_v \cup (V_2)_v$ . But this is absurd because the germ of an analytic space at a smooth point is irreducible. This boils down to the fact that the local ring of a smooth point of an analytic space is an integral domain, which is clear since it is a a ring of convergent power series $\mathbb C \{z_1,\ldots, z_n\}$.

By the way, judging from your notation, I suppose you extracted this question from Griffiths-Harris. I find their treatment a little cavalier, since indeed they give no explanation at all for this assertion.

If you want full details, I recommend the brothers Kaup's book Holomorphic Functions of Several Variables (de Gruyter Studies in Mathematics 3), where they prove that a reduced complex space is irreducible iff its smooth points form a connected open subset (49.7 Corollary, page 194).

And, last but not least, happy New Year to you and all our friends of MathOverflow !

Dear unknown, here is a sketch of proof of your question ( which I have modified to make it more accurate, as explained in my comments to your original post .)

Statement If $V=V_1 \cup V_2$ with $V_1, V_2$ irreducible and distinct from $V$, then the intersection $V_1 \cap V_2$ consists of singular points of $V$.

Sketch of proof Suppose there is a point $v\in V_1\cap V_2$ which is holomorphically non singular on $V$, i.e. holomorphically smooth. Then the germ of analytic space $V_v$ would have a decomposition $V_v=(V_1)_v \cup (V_2)_v$ . But this is absurd because the germ of an analytic space at a smooth point is irreducible. This boils down to the fact that the local ring of a smooth point of an analytic space is an integral domain, which is clear since it is a a ring of convergent power series $\mathbb C \{z_1,\ldots, z_n\}$.

By the way, judging from your notation, I suppose you extracted this question from Griffiths-Harris. I find their treatment a little cavalier  , since indeed they give no explanation at all for their assertion, which is actually not quite correct, as explained in my comments to your question.

If you want full and details, I recommend the brothers Kaup's book Holomorphic Functions of Several Variables (de Gruyter Studies in Mathematics 3), where they prove that a reduced complex space is irreducible iff its smooth points form a connected open subset (49.7 Corollary, page 194).

And, last but not least, happy New Year to you and all our friends of MathOverflow !