We have two spaces. $X=S^1\times S^1$ without $J\times J$ where $J$ is short curve on $S^1$
We obtain second space $Y$ by gluing together two spaces like $X$ along their holes (I mean those obtained by removing $J\times J$).
I am not sure how to do it. Visually I can see that we can deform $X$ so as to obtain homotopically equivalent $S^1\vee S^1$ therefore its fundamental group is free group on two generators, but I suspect that this time I should use something stronger than such deliberations. Is there chance that is somehow follows from Seifert–van Kampen theorem or something related to it?