This is motivated by reading the notion "locally square integrable local martingale". Parsing that I arrive at a process $X$ for which two sequences of stopping times $(\sigma_n)$ and $(\tau_n)$ exist, for which $\sigma_n\uparrow \infty$ and $\tau_n\uparrow\infty$ and for which $X^{\sigma_n}$ is square integrable and $X^{\tau_n}$ is a martingale. Can we find a single localizing sequence $(\rho_n)$ for which $X^{\rho_n}$ is both a martingale and square integrable?
More generally, given multiple local properties just as above for a process $X$, can we find a single localizing sequence $(\rho_n)$ for which the stopped processes $X^{\rho_n}$ has all those properties (and additionally is a bounded process provided the process is sufficiently semicontinuous)?
To answer your first question: Yes, you can. But not in the "natural" way to define $$\rho_n := \sigma_n \wedge \tau_n$$ because square integrability is not guaranteed for the stopped process in general. But it is if you have square integrability of the maximum process $$X^*_t := \sup_{s\le t} |X_t|$$ Then it follows for any stopping time $\rho$ that $$\begin{align*} E[X^2_{t \wedge \rho}] &= E[X^2_t1_{t\le \rho}] + E[X^2_\rho1_{t > \rho}] \\ &\le E[X^2_t] + E[X^2_\rho] \\ &= E[X_t^2] + E[X_t^*] < +\infty\end{align*}$$
So then square integrability becomes stables under stopping.
Fortunately martingale property is stable under stopping so if $M$ is a martingale and $\rho$ an almost sure finite stopping time we have that $M^\rho$ is a martingale as well. This leads to the useful fact that if we have a localizing sequence $(\tau_n)$ s.t. $X^{\tau_n}$ is a martingale and we define $$\sigma_n = \inf \{t \ge 0\;|\; |M_t| > n\}$$ then $\sigma_n$ is a stopping time and with $$\rho_n := \tau_n \wedge \sigma_n$$ we get that $M^{\rho_n}$ is a bounded martingale hence square integrable.
So, to summarize: For properties which are stable under stopping you easily can combine all related stopping time by considering the infimum of all stopping times.
For the other ones you still can construct a stopping time so that you get a bounded martingale and most of the properties hold for bounded martingales.
If you have property which is neither stable under stopping nor valid for bounded martingales you have to find another way.