It is terribly hard to define sentience. These are all based on my feelings regarding the word, so you may have different opinions.
What would these individuals "do" to show their sentience.
The most exciting phrase to hear in science, the one that heralds new
discoveries, is not 'Eureka!' but 'That's funny...' - Isaac Assimov
When a new sentient species forms, it isn't treated as a species at first. It'd be treated as a sub-species until human scientists decide they are different enough to be treated as independent species. Until then, it would just be a group that has a curious tendency of "getting lucky" more often than their brethren. The first uses of sentience would not be huge neon-signs saying "I'm intelligent" like speech, writing, or tool making. The first uses would be improvements on the existing models the non-sentient base species was using. If this was a predator species, you would notice that a few individuals had a curious tendency to be very effective at predicting prey, forcing them into situations which have no win scenario for the prey. If this was a prey animal, you would notice that a few members of the group were curiously absent when attacks occurred, or had some other curious way of not becoming dinner.
What would be the core debates of the original sentient species?
I believe that the biggest problem that humanity faces is an ego
sensitivity to finding out whether one is right or wrong and
identifying what one's strengths and weaknesses are. - Ray Dalio
If you have only one species that is sentient, they will likely have fine tuned their definitions of sentient to their own needs. There is a great need in a culture to declare "sentient" as a lower bar for society, but less of a need to define a line in the sand to herald the coming of a new sentient species. Accordingly, their definition of sentience is going to be very customized to the culture and physiology of their own species. For example, a human like culture may define "sentience" as the ability to read and write, or do arithmetic. We often draw the line for sentience around "complex language," which itself is a very fuzzy concept.
How well a culture can adapt their concept of sentience to reflect the new evolving reality will depend heavily on how comfortable their culture is at the time of the revelation.
The fear response
Under the influence of fear, which always leads men to take a
pessimistic view of things, they magnified their enemies' resources,
and minimized their own. - Livy
If a culture is strained due to existing conflict (internal or external), it will look at the new species with eyes that judge rapidly. It does not have the luxury of working with the new species to see what happens. This will likely result in either annihilation or slavery for the new species. The older species will look at ways to use the new species as a tool to solve it existing problems. If there is a good fit for this, the older species will naturally treat the younger species as inferior in order to keep them pigeonholed in this tool role.
On the other hand, if the older species cannot find a use for the new species, it is a liability. At any point in time, when the older species is trying to balance solutions to its existing conflict, the new species may come barging in and disrupt everything. Accordingly, the older species will make sure this cannot come to pass. Walls may be sufficient for a while, but eventually the response will have to become more violent (because the older species is too busy to take the time to be non-violent). This may be "pain therapy" for the young species, or it may result in a straightforward elimination of the entire group if fear runs high enough. There will be no incorporation.
The acceptance response
Lots of people talk to animals...Not very many listen, though...That’s
the problem. - Benjamin Hoff
If the society has more room to spare, it will have more ability to try to understand this new mind. This will include things like redefining "sentience" to a more useful definition. This side of the argument is easier to make, because we have a lot of historical evidence to work with. Historically, we have had to accept other groups on many occasions, and those can be used to analyze what might happen with an inter-species acceptance.
Throughout history, there have been constant castes and factions in humanity. Some of the time, they manage to get along well enough to form a larger community. The ultimate example of this (for Americans at least) is the freeing of American slaves in the civil war era. We were forced to accept that there was an alien culture living with us that we could no longer simply call "property."
There could also be patterns similar to dog training. As we interact with this other culture, we may find niches in our society where they work better than we do. This is similar to the "tool" argument for the fearful response, but it isn't quite the same. This would be more like a working-dog situation. Anyone who works with dogs knows they need to be treated with the greatest of respect if you want them to function at peak capacity in a working setting.
What could the new species do?
Wait for opportunity before moving, watch for changes
Create opportunity by following the opponent's force - Wu Jianquan
This was not a question you asked, but it seems reasonable to look at both sides of the situation. After all, both sides are sentient.
The new species is eventually going to become aware of the Other Mind of the older species. Once it becomes aware of this, it will need to act accordingly. The actions of the new species can help or hurt their chances substantially.
The most important character trait the young species will need is patience. If they attempt to rapidly force the older species to accept them, it removes degrees of freedom that the older species needs. A species may be able to accept your new species in 50 years, but if you do something which forces their hand in 5 years, they are more likely to respond violently.
The biggest issue will be sudden events which show an unusual level of aptitude. If a young species suddenly tries to communicate with the older species, the older species will have to deal with that communication on a short order. This may have any number of effects. If the older species is strained, it will likely result in a summary execution of your new species. If they are comfortable, it may give the new species some clout. If the old species is starting to wonder what the next step in their path will be, this could be a great thing heralding rapid acceptance by the older species.
The key takeaway from all of that is that the result of a fast moving action is completely in the realm of the older species. The new species can't really decide what the result is, only when it will occur.
If the younger species is more patient, the older species will naturally flex around them at the older species' pace. As long as nothing threatens the younger species (like habitat changes), it is better to move slowly and see what happens. In this situation, the younger species can't decide when changes will occur, but they will find they have much more control over facets of the result.