I'm not sure that I fully understand the question such that my answer will be of any use, but regardless of the measure whether this be units or money, surely if there is a distribution or frequency observed according to some other variable, such as different cities in that maybe there is a steady rise of capital and labour units with increasing socioeconomic status using a hypothetical example for the sake of trying to explain myself, would you not get the same significance type value regardless of the unit, in that the trend will not alter. I don't know if this makes sense or is correct, I am just guessing really.    if you go to my website to the favourite links page and see the electronic statistical textbook, maybe that can help. http://sites.google.com/site/deborahhilton/  

In practice, there isn't really a difference between units and monetary values. Usually, there are data on L in units (e.g. fte or hours worked) but not on K for which you would use the total value of the capital stock. When you measure a production function such as yours with longitudinal data and data for different countries, you must adjust your monetary values for inflation (e.g. using a GDP deflator or specific deflators for investments, labour input, output, etc.) and you must adjust for international differences in price levels using purchasing power parities. The resulting figures amount to indices of the volume of inputs and outputs, which brings you back to your units.

For more information, check out the publications of the Groningen Growth and Development Centre, specifically those of Bart van Ark and Marcel Timmer, and have a look at the total economy database of the Conference Board.

You might also want to consider expanding your model to include knowledge as a separate production factor.

Measurement in physical unit gives real growth picture and monetary unit gives nominal growth picture.

You want to be able to distinguish between physical and price movements. In empirical application, the way the data is available might need some massaging, and the specification of the model must be done with care.  I attach a page from P. Dhrymes, an authority, on what is involve.

Increase/decrease in production in terms of monetary unit is different from real units; there is price effect in former measurement, for instance money-value of production would increase due to rise in price even though there is no change in level of production. C-D production function can be expressed in both terms if data are availble.

If your data are time-series then the monetary measurements are often more likely to be trended (by inflation) and thus more at risk of generating a spurious regression.

Both Vince Daly and Edwin HorKlings added some helpful points that seemed a bit more macro in nature, so I thought I would add something on the micro side, especially since you have areas of expertise in both applied micro and applied macro.

One advantage of using units such as the number of workers and the number of machines along with the amount of output is the ease of interpretation and use.  This might be a firm which has a multi-plant operation, especially if it uses more labor intensive production processes at some of the plants than others. Alternatively, it might be a time series example if the same plant has used different processes over time.  Both have interesting complications that should be considered; however, staying focused on your question suggests one of the pros is the ease of calculating the marginal products of labor and capital MPL and MPK respectively. 

Knowing these and the prices of labor and capital (i.e., PL and PK),  allows us to find the MPL / PL  compared with the MPK / Pk and if they differ, it suggests some adjustments to reduce costs and increase profitability might be in order.  Although it may be mathematically possible to manipulate monetary values for the inputs and output and get similar results, the explanation is a bit convoluted. Just try explaining monetary values in this context and you will see what I mean.

In addition, the scale factor may come into play, which is mentioned in Lall Ramrattan's link to Dhrymes. That one page has a powerful message that too important to ignore.

The second set of comments of Arjun Pangannavar come into play about monetary values.  Remember, physical units do not need to be deflated; however, all of this assumes the units of labor, capital, and output do not vary in terms of quality. It is like assuming the workers are identical twins and beyond working with identical machines producing uniform-quality identical products.  The more we deviate from this in the real world, the more challenging it is to be assured our results are meaningful. Regardless, we do the best we can and you are fortunate to have the data both ways.  I hope this helps.