Primal vs dual decomposition methods
I don't think one can make a blanket statement on whether primal decomposition methods or dual decomposition methods are more efficient. The efficiency is dictated by the structure of the problem; primal decomposition and dual decomposition are ways of exploiting this structure to develop efficient algorithms.

The key aspect in making a choice between these methods is the kind of parametrization the master brings about on the slaves. Fo e.g., sometimes it is convenient (in terms of theory or practice) to have the parametrization in the objective function of the slave problems instead of having it in the constraints. If the parametrization happens to appear in the constraints (and it has suitable structure), one can try to dualize the slave problem to get in the objective.

In the slides you have linked, the effort seems to be in exploiting the structure for parallelization of the solution of subproblems. For parallelization one would require subproblems that are independent of each other. If there is a natural separability in the constraints, dual variables can be used to parametrize the problem and thereby create such subproblems.

Edit: I repeat once again that structure is important in determining the choice of the method. I am still not completely clear on the structure afforded by the problems you have in mind. Based on the information given, this is my best attempt at answering your question. 

From the examples you have added, the problems appear to be what can be called "nearly separable". In essence, these problems appear to be variants of the problem $\min \{\sum_{i=1}^n f_i(x_i) : (x_1,...,x_n) \in K\}$, where $K$ is a constraint that has a separable structure. For example, the constraint $(x_1,...,x_n) \in K$ may be $\sum_ix_i =b$, where $b$ is a vector of appropriate size. By separable structure I mean that the optimal values of variables $x_1,...,x_n$ can be chosen independently of each other. If one looks at the problem as a whole, the objective functions are decoupled in individual variables, but because of the coupling in $K$ these variables cannot be independently optimized. However because this coupling in $K$ is mild, one can still salvage separability. Dual decomposition lets you separate the variables when they are coupled in the constraints. If the variables were coupled in the objective, one may have to employ primal decomposition (as in Boyd's slides). One may of course try to reformulate problems to bring them in some or the other form that one is comfortable using.

To demonstrate separation by dual decomposition, define the Lagrangian $L(x,\lambda) =\sum_i f_i(x_i) + \lambda^T(\sum_i x_i)$. If $(x,\lambda)$ is a saddle point of $L$, then $x$ solves the original problem. A dual decomposition method would for each value of $\lambda$, minimize $L(x,\lambda)$ over $x$, altering $\lambda$ until a saddle point is found. The minimization in $x$ is easier, in particular separable and parallelizable.