As I explain in the updated question I don't think this has a useful general answer but I'd be very happy to understand why I'm wrong so any ideas are welcome. Thank you very much for the interest!

Exactly. If one restricts $g, \varphi \in H^{1 / 2} ( \Gamma)$ to the subsets $_i$ and calls these functions $g_i, \varphi_i$, then $\langle g, \varphi \rangle_{H^{- 1 / 2} ( \Gamma)} = \sum_{i, j} \langle g_i, \varphi_j \rangle_{H^{- 1 / 2} ( \Gamma)}$ and this sum may be split as $\sum_i ( g_i, \varphi_i)_{H^{1 / 2} ( \Gamma_i)} + \sum_{i \neq j} \langle g_i, \varphi_j \rangle_{H^{- 1 / 2} ( \Gamma)}$, but the mixed terms are not representable using the scalar product in $H^{1 / 2} ( \Gamma_i)$.

But $n$ is the dimension of the space and the integral is on the boundary so that's one dimension less...

Hi, thanks for your interest. I guess you mean the scalar product, right? See for instance these lecture notes, proposition 4.4: science.unitn.it/~visintin/Sobolev2011.pdf . I used this scalar product because the one that came immediately to mind to me from the Gagliardo norm was more complicated, involving differences of the arguments. I haven't actually checked whether they are equivalent, but it seems plausible enough. I've also left out the details about the surface measures using the charts for the boundary $\Gamma$, etc., thinking that was non-essential.