Mixture probability depends on the sample A mixture of two distributions has density which is the weighted sum of the components:
$$f_{mix}(x) = p f_{1}(x) + (1-p) f_{2}(x).$$
What if the mixture weight is allowed to vary with the sample point?
$$f_{mix}(x) = p(x) f_{1}(x) + (1-p(x)) f_{2}(x).$$
That is, you need to know the sample $x$ before you can determine what mixture it came from. It's unclear how to draw from this distribution, and in fact this is not a distribution, as $f_{mix}$ will not in general sum to 1.
Nonetheless it seems interesting. For example, I can create a skewed distribution by "switching" from one normal distribution the other, both centered on 0:
$$f_{mix}(x) = \frac{1}{Z} \left[\text{sigmoid}(x) N(x; \sigma_{1}) + (1-\text{sigmoid}(x))N(x; \sigma_{2})\right].$$
Is there a name for this idea? Can this be realized as a physical model, perhaps under some conditions?

That is not a distribution at all --- the purported density function does not integrate to one.  In general, when you have a proper mixture distribution (i.e., one that uses fixed weights) you can rest assured that:
$$\begin{align}
\int \limits_\mathbb{R} f_\text{mix}(x) \ dx
&= \int \limits_\mathbb{R} [ p f_1(x) + (1-p) f_2(x) ] \ dx \\[6pt]
&= p \int \limits_\mathbb{R} f_1(x) \ dx + (1-p) \int \limits_\mathbb{R} f_2(x) \ dx \\[6pt]
&= p + 1-(p) =1. \\[6pt]
\end{align}$$
However, using your proposed change you get:
$$\begin{align}
\int \limits_\mathbb{R} f_\text{improper mix}(x) \ dx
&= \int \limits_\mathbb{R} [ p(x) f_1(x) + (1-p(x)) f_2(x) ] \ dx \\[6pt]
&= \int \limits_\mathbb{R} p(x) f_1(x) \ dx + \int \limits_\mathbb{R} (1-p(x)) f_2(x) \ dx, \\[6pt]
\end{align}$$
which does not equal one in general.  What you are proposing here would require you to add a scaling constant to your density to make it integrate to one.  That just leads you back to the same level of generality you have when you start by saying that you can form a density function by scaling any non-negative function with a finite integral.  In view of this, it is not really clear what utility your proposed method has.
