The form at the bottom of this page allows you to estimate the number of intelligent civilizations within a given distance of Earth.
This is quite a bit different from the Drake equations, although it retains some of the parameters. The Drake equation is a very general purpose approach to counting aliens. This form, which I call the Local Equation, takes out some of the parameters of the Drake equation because it is estimating only the number of intelligent civilizations on nearby stars.
The main difference in the Local Equation and the Drake equation is that the stars within a few thousand light years of earth are largely of the same ages and types. There is some variation, but most stars in this part of the galaxy seem to be related to each other and many have similar spectral types, ages, and, we might assume, similar types of planets.
Local density of stars is about 0.001 stars per cubic light-year. This does not include many stars too faint to see, but it gives a good rule of thumb. The types of stars in a group of 1,000 would be divided as follows:
1 star of spectral type B
7 stars of spectral type A
27 stars of spectral type F
65 stars of spectral type G
126 stars of spectral type K
774 stars of spectral type M
The Sun is type G, but it is speculated that almost any type of star can have a planet. Type M is a Red star, which may have undergone changes that would have killed off planets. Types F, G, and K are nearer the size of the Sun, some are hotter and some are colder, but all are "of an age" and may be able to support life.
As many as 60% of all stars are double stars, and there have been some calculations that show that double stars might cause problems for planets. Planets might not last long in some double star systems or they may occasionally pass very close to one or more of the stars. Other calculations show that planets that might support life are possible for double stars, but might not be as common as single stars.
A good estimate might be that 10% of the stars in this neck of the woods are very similar. They are similar in spectral type to the Sun, they are the close to the same age, and they are about the same size. It would be strange if they did not have similar planets in a similar layout to that of the Sun.
So starting out we know the number of stars in this region of the galaxy that probably have planets and could support life.
There is are a few further factors. Since it took humans about 6 billion years to come to the fore, can we guess that it might take other civilizations 6 billion years to appear? What would be the window? Could civilizations have started as early as a billion years earlier on precocious planets. Could slower planets need an extra billion years? This is added as a parameter to the form.
The form asks how long a civilization will last, which can be from 1000 years to 100 million years. Older civilizations would have to be very stable to last that long. I personally don't think a civilization can last longer than a thousand years before becoming something else and not likely to communicate.
Another question is how often to intelligent civilizations arise on a given planet. Does a new one pop up every million years or even less. If a civilization only lasts 1,000 years, when does the next one spring up?
One last question would be how many civilizations would seek to contact us or respond if we try to contact them?