Are short-let platforms really contributing to rising rent prices? Short-term rent services such as AirBnB are often criticized for eroding the communities by increasing rent prices for locals:


  ... as short-term letting transfers much-needed housing from the residential sector into the tourist economy, it [...] contributes to rising rent levels ...


While it makes sense that with less houses available for long-term rent, the rent price goes up, I would expect the effect of short-term rents to be insignificant, as most properties proposed on AirBnB are not available for long-term rents anyway.

Are there any studies which managed to quantify how big this effect really is?
Yes an example of such study is this Sheppard, S., &amp; Udell, A. (2016). Do Airbnb properties affect house prices. Williams College Department of Economics Working Papers, 3.

This is their conclusion:


  We find that in New York City, the impacts appear to be that an increase in localized Airbnb availability is associated with an increase in property values. In our hedonic model estimates, a doubling of Airbnb listings is associated with increases of 6% to 11% in house values, ceteris paribus. Using a difference-in- difference approach produces an even larger estimated impact, suggesting that properties that are subject to the Airbnb treatment increase in value by about 31%. Rough calculations based on average property values, average Airbnb rentals, and an assumption that potential income streams will be fully capitalized produces an intermediate estimate of about 17.7%.
  
  While our results might be taken as supporting critics of Airbnb who complain that the firm’s services act to increase house prices and diminish housing affordability, we want to stress that this conclusion may be unwarranted. A service that increases house prices (such as improved police protection, making better local schools available to residents, or providing more and better public parks) need not diminish community well-being.
  Public policies that reduce house prices in pursuit of housing affordability by diminishing the efficiency with which an owner can make use of his or her property are unlikely to be welfare-improving, in the same way as a city that creates “affordable” housing by encouraging more crime hardly seems desirable.
  
  Evaluating the welfare consequences of Airbnb, and hence the appropriateness of any regulatory action to limit use of Airbnb services, requires deeper analysis than we have provided here and much deeper analysis than appears to have been undertaken to date.