This is a very good and hard question. Based on your question, I think you probably already know something about xeno transplantation, the idea of replacing damaged human organs with a healthy animal organ. Your question asks if we can use animals (like pigs) to grow human organs, which can then be used to replace a damaged version of a human organ. The goals are similar, and the problem that you brought up (tissue rejection due to immune system attack) is the main problem in all of this area of research. I'll tell you about three areas of current research and how scientists are hoping to overcome the problems.

Keep in mind that the main problems with all of these methods of organ transplantation are tissue rejection and disease transmission. In tissue rejection, the immune system of the patient sees the new organ as foreign and creates antibodies which attach and flag the cells as invaders, causing white blood cells to attack the transplanted tissue. This massive immune reaction is often fatal. Putting anything foreign into the body also creates problems because the new tissues may carry diseases which the body is not prepared for, especially when the body is under stress from the drugs associated with transplant operations.

Let's start with your specific idea: using cloning to grow human organs inside of other mammals. You may have heard of stem cells. I won't go into the biology of stem cells here. Suffice it to say that stem cells are completely un-specialized and can be trained to become any type of tissue: skin, liver, heart, muscle, stomach, etc. In theory I can grow a stem cell colony in a Petri dish and, using hormones, cause the colony to grow into any tissue I want. Most cells, once they are specialized, cannot be reset thus if I was growing skin cells in a Petri dish I cannot add anything to make them grow into a heart. The best human stem cells are the first few cells of an embryo. There are many moral problems with using these cells because it kills the embryo. There are stem cells found in the marrow of everyone's bones. Although they are harder to use, they are more powerful because harvesting them doesn't hurt anyone much.

One set of researchers has developed a way to put these marrow stem cells from a human into a baby sheep still in its mothers womb. When the sheep is born, all of its body parts will have some characteristics of the human who donated the marrow stem cells. In theory, if the liver of the sheep were transplanted into the human it would not be rejected because some portion of the cells in the liver are derived directly from that person. When the organ is transplanted, the immune system is much less likely to produce antibodies to the cells of the organ because it seems to recognize some percentage of them as being native.

The most successful attempts at this have produced sheep whose organs are about 15% human cells. Researchers say it will be at least 15 years before this technology can be tested in human transplant patients.

Another area of study is human cloning.

In this method, a person's DNA would be put into a fertilized human egg which would produce stem cells matching the DNA of the donor. These stem cells could be grown up into an organ which would exactly match the donor. This dramatically reduces the likelihood that the patients immune system would attack the transplanted organ.

A third area of study is called xeno transplantation. This is the most likely one to actually begin saving people in the next twenty years. Basically the idea is to take an animal which produces lots of babies quickly and change its DNA to make its organs look more human. Pigs work well for this. You can grow pigs in a sterile environment (reduces the risk of infection) and then place their organs into a sick human. The problem is tricking the humans immune system into accepting the organ. One success has been to replace certain pig genes with human counterparts. These genes are the ones that make proteins which alert the human body to the foreign invader. In theory, these organs look part pig and part human and very weird to the immune system, but don't look scary enough to cause the immune system to go on full alert.
For more information, check out these links:
Cloning
Xenotransplants