Let $F_t=A_tK_t^\alpha L_t^{1-\alpha}$ be the production function with two parameters.
In regression, we know the firm level $F_t$, $K_t$, and $L_t$. We want to estimate $A_t$ and $\alpha$.
I've heard that, for the same industry, $\alpha$ is the same for all the firms. $\alpha$ is also usually assumed to be the same across the time period.
However, the technology parameter can shift across the firms and times.
I am not sure if I am correct on this.
You can estimate it in multiple ways. If you have panel data then you could first log linearize the expression giving you:
$$ln(F_{it})=ln(A_{it}) + \alpha ln(K_{ti}) + (1-\alpha) ln(L_{it}) $$
Which could be estimated in multiple ways. You could estimate it as a pooled cross-section where the technology would be assumed constant across the time and same for all firms (lower case variables are in logs, and betas here give you the estimates of exponents from original specification):
$$f_{it}=a+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$
You could run firm fixed effects specification where you allow every firm to have its own technology but technology is constant across the time:
$$f_{it}=a_i+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$
You could assume that it’s the same for all firms but different across time:
$$f_{it}=a_t+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$
Or you could also have a specification where you would assume that technology can be different across firms and also change in time (but the change in time has to be the same for all firms)
$$f_{it}=a_t+a_i+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$
If you would also want to allow for technology to change differently across firms you could employ some multi equation estimation where you estimate the parameters for each firm separately.
So really depending on what you think is the most appropriate for your dataset you could make different assumptions on this. For example if it’s an industry that relies on basic research so there are no intellectual property rights - it might be appropriate to assume all firms have the same technology which can change in time. If it’s some industry where technological progress is unlikely - let’s say some services like barbers then the best assumption might be to have constant technology for all firms and times etc.
