I can't see this being an issue. Parents don't pass on changes that happened during their lifetime (organ transplants, amputations, scars, etc.) to their kids (look up "Lamarckism" for the discredited theory about this). The only way this could affect future offspring is if you were somehow to manipulate the reproductive cells (ovaries and sperm). At least, that's my understanding -- IANAD.
That is correct. You can change a human in all kinds of ways but unless you've changed the DNA in the sperms or ova (or the cells that generate sperms and ova) it's not going to be passed on to the offspring.
Growing an extra organ inside a human isn't going to change their DNA. Even if someone genetically engineered cells anywhere in your body apart from the specific ones in their testes for sperm making, the change won't be inherited. In women a woman's ova are already formed before she's born, they wait in the ovaries until her menstrual cycle starts before popping out, one by one, each month, so you can only create inheritable change by altering the actual ova. But organ transplants or doing weird stuff like grafting on an extra arm onto someone isn't going to change the DNA, even locally, so there's no chance of it being inherited.
A transplanted organ still has the donor's DNA so unless the donor is your identical twin, your immune system rejects it as foreign/non-you cells and sets out to destroy it (drugs to suppress the immune system are needed to prevent this in people who've had transplants, leaving them vulnerable to illness). If they are growing organs with someone else's DNA in them, they will need to stop the immune system from destroying it. If the new organ are from the person's own DNA, this would avoid the problem of the immune system rejecting and destroying it.
Also, what do you mean by "new specialist organ" - all organs are specialist at what they do. Your heart's specialised for pumping blood round your body, your stomach's specialised for digesting food, etc.
If you're looking for a technology to make organs start to grow in places in the human body where they're not supposed to be - this sounds a bit like a teratoma - a type of growth that has tissues of other organs in it (note to the squeamish: don't google teratomas because they can be scary and gruesome) I can't remember if they're technically classified as cancer or not, but they're along the same lines as cancer - cells doing the wrong thing in the body and growing into something they're not supposed to be.
A quick summary of the science - all your cells with a nucleus have your entire genome in them* but most of the DNA is switched off. The code for growing eyeballs is switched off in your liver cells - but the relevant DNA is still there. With a teratoma, it appears to be that the wrong bits of DNA are switched on when they're supposed to be switched off. The result is a growth that contains bits of tissues from completely the wrong parts of the body.
*some exceptions like red blood cells don't contain any DNA at all and don't have a nucleus
So, if you want to grow an eyeball inside someone's liver, you just need to know how to switch the liver cell DNA instructions off and switch on the "grow an eyeball" instructions instead. As long as your story's not hard science fiction, you can use a bit of handwavium to explain how they switch these genes on and off. I think you may need to start with a stem cell to grow a whole new organ but teratomas form somehow and they grow inside of adults, so there may be a way to make this work with developed, adult cells.... I don't know how much handwavium you want to have in your explanation.
Note that this only works for organs that the DNA already codes for. If by "new specialist organ" you mean some kind of engineered organ that's new and different to existing human organs, this method won't work. Also, because it's the person's own DNA, if you were to grow an eyeball in their liver, if they've got brown eyes the liver eyeball would also have brown eyes. It wouldn't be able to see though, as it won't be connected to the optic nerve or the visual cortex of the brain.
Stephen King had something that was somewhat along these lines in his novel The Dark Half... was a teratoma type thing. The condition foetus in foetu (where someone has an embryonic identical twin inside them) is possibly a kind of teratoma rather than a twin that's been there since early foetal development. Often they are not complete foetuses but random assortments of tissues that are well enough organised to resemble embryonic structres of the human body. The Dark Half had some kind of connection with foetus in foetu but I can't remember the details because I read it more than 20 years ago.
If you're doing some kind of something to the DNA inside cells to switch on and off different bits of DNA this would increase the risk of cancer if nearby cells' DNA is accidentally changed as well. Whatever they do to the DNA to change which bits are switched on would have to be very localised and probably done through microsurgery. Maybe they can do that to the DNA in an individual nucleus in an individual cell and make it start to grow into the the wrong kind of cell. You may have to do something to make it become like a stem cell first then switch on the right bits of DNA to make it grow into the desired organ.
Another way to do this would be to transplant stem cells - pre-programmed (for want of a better term) to be like the embryonic cells that will grow into the desired organ, then transplanted into body tissues where they'll get enough blood supply to grow.
Bear in mind also that if you are growing new organs like this that the organs will start of embryo sized, then foetus sized, then baby sized, then child sized etc and also that their development will probably be messed up by the fact they're not where they're supposed to be and not doing what they're supposed to do (a heart grown in this way won't be part of a circulatory system being filled with blood, it'd be empty and attached to the wrong things.) It's up to you how far you want to go with the technology to solve these issues, for example whether they have the technology to accelerate growth (this might affect the person they're growing the organ inside of and make bits of them age faster) and they may need to do surgery to correct any developmental issues with the organ. Presumably they're going to transplant it somewhere useful at some point? At this point I'm concerned I've completely missed the point about where you're going with this and have been typing at 90 words per minute about teratomas and growing eyeballs in livers for no reason.... it was an interesting thought experiment anyway so I hope some of it's been useful.
DISCLAIMER: IMO the above is not scientific enough to fly in hard science fiction. The stuff about how DNA and cells work is correct to my knowledge, but the technology of manipulating DNA in this way is speculative and I can't be specific about how they'd do these things. I'm not aware of such technologies existing. But the technology of growing stuff like this in the lab is getting pretty advanced and my knowledge of it is not up to date.