I was fortunate enough to spend the past week at a workshop held at the Green Bank Observatory (GBO), so I can give you a partial answer based on how the Breakthrough Initiatives affects it in particular.
The GBO actually operates a selection of telescopes, of which the most widely-known (and largest) is the Green Bank Telescope (GBT). The GBT is the main instrument used by Breakthrough Listen at the observatory. During the second half of 2016, the GBT was oversubscribed by a factor of 2.6, meaning that only about one in every 2.6 proposals was approved outright (“filler” proposals will be accepted if short periods of time are available). This doesn’t mean that for every hour of telescope time, 2.6 hours worth of observations were proposed, although the ratio is close (2.3), but it does mean that the GBT is very much in demand - and those numbers have increased this year. The telescope is not in use on only two days annually, Thanksgiving and Christmas, although there are blocks set out each week for maintenance.
The answer to your first question, then, is yes - there is approximately indeed a zero-sum redistribution, because the telescope is extraordinarily busy. The actual data-gathering proportion may be slightly different because overhead times vary by the type of observations, and things like changing receivers or pointing at a different source do take time. However, it is true that Breakthrough Listen, like any other set of observations, takes time away from other scientists - about 20% of the GBT’s annual total observing time.
(As an aside, Breakthrough Listen operates a “backend” in the GBO computer room - a set of GPUs that analyze incoming data. There are plenty of backends for various receivers and instruments, and they do take up space - and power. We were told that the room the Breakthrough Listen computers are in wasn’t actually designed to hold that many computers, which did cause some initial cooling issues (which were quickly rectified) when these new units were installed. So it’s important to realize that the project takes up more resources than just observing time!)
The financial system at Green Bank is actually a little complicated. In 2012, it was recommended that the GBT and the Long Baseline Array be removed from the NRAO, and this was implemented in 2016. In some ways, the transition is still happening - much official GBO information is still on the NRAO’s website - but the relevant point here is that in the 2017, the GBO only received about 66% of its funds from the National Science Foundation (NSF). The current agreement is that the NSF will gradually defund the GBO over the next few years.
This is why Breakthrough Listen is so financially crucial, and why it is getting so much publicity at the GBO. Without sources (and currently, without non-governmental sources) of funding, the observatory simply cannot operate. That said, the GBO is not beholden to people like Yuri Milner (the man behind the Breakthrough Initiatives) who have specific projects in mind - and indeed, that type of donor is quite the outlier. It does seek out donors and philanthropists who are willing to give money to simply ensure that the facility keeps going.1 If it can be guaranteed that these sources will continue giving without setting limits on what types of observing should be done, then things can progress as close to normal as possible.
The exact details of funding for the GBO are not (as far as I know) public, and I don’t want to speculate too much. However, I can say that the observatory is almost certainly not going to push ordinary astronomers out of the way for the sake of big projects - in the short-term. Long-term funding is uncertain, true, but that does not mean that it will be renamed the Breakthrough Listen Observatory any time soon. The number of people who have operated the GBT number in the thousands, and they don’t pay to use it. It was built using public funds and has operated largely using public funds for quite some time now. I think the GBO intends to keep operating like this, and will not turn to the alternative - subject to the interests of only a select few projects - unless matters become extraordinarily desperate. It’s possible, but not something to be immediately worried about.
1 One day last week, we noticed a helicopter landing at the airstrip. It turns out that the passenger was a VIP who was given a full tour of the telescope, guided by several observatory higher-ups. He was, as you can image, a prospective donor.