There are two issues here: (1) Can Jews eat Halal? (2) Can Muslims eat Kosher.

Of course, we are assuming that the Jews or Muslims are following their religious dietary laws to the letter. Obviously, Jews and Muslims are physically capable of ingesting food products prepared according to the other’s dietary requirements, so the question is really about the law.

(1) Can Jews eat Halal?

Concerning Meats

As a number of the other Answerers have noted, especially Meir Lipnick, the process for kosher slaughter of meat products has many requirements that halal slaughter or Zabiha (ذبيحة) does not. Accordingly, a Jew who keeps kosher will not eat halal meat.

Concerning Vegetarian Dishes

While it would appear that a halal vegetarian or vegan restaurant (that serves no egg, fish, or meat products) would be satisfactory, the laws of Kashrut would prevent consuming the food here as well. There are a number of issues that could arise:

1. Spices are often prepared from living creatures which are themselves not kosher. A perfect example is the Indonesian spice/sauce called sambal which uses crushed shrimp in its creation.
2. Often times a kitchen has not been properly checked to make sure that there are absolutely no non-kosher items.
3. Vegetables may not be adequately cleaned to remove all of the bugs that are cultivated along with the vegetables.
4. Under Kashrut, cheeses need to be produced without animal rennet, which is not required in Halal. ( animal rennet used has to be halal slaughtered as well to be halal )
5. Some Jews follow a Rabbinical command that cooked foods must be prepared by a Kosher Jew.

(2) Can Muslims eat Kosher?

It depends on the particular issue you are raising. However, most Muslims feel that kosher food is much closer to halal and can be acceptable as compared to non-kosher and non-halal certified products. In general, Jewish dietary restrictions are more rigorous than Islamic ones, so this is why kosher is usually acceptable. There are two restrictions and issues, though.

Concerning Alcohol

Kahrut permits alcohol while Islamic Dietary Laws forbid it. As a result, items cooked or prepared with alcohol, such as coq au vin or tiramisu, may be completely kosher, but are haram because they include alcohol.

Concerning Meat (land animals and birds)

There are some debates among Muslims as to whether Jewish blessings during the slaughter of an animal is sufficient to qualify as tasmiyah (تسمية) (blessing God in an Islamic context during slaughter). The majority opinion is that since Jews bless the same God as the Muslims for the same purpose, the Jewish blessings count as tasmiyah. There is a minority opinion that the failure by Jewish butchers to say specifically “Bismillah ar-Rahman ar-Rahim” (بسم الله الرحمن الرحيم) when slaughtering an animal and/or to not bless every single animal, as opposed to entire queues of animals, is sufficient to bar the consumption of kosher meat. This arises primarily as an issue in the Hanafi school of Sunni Islam. No other school has an issue here. There is no other issue for a Muslim eating kosher meat.

Concerning Seafood

In Sunni and Ibadi Islam, there are no restrictions as regards to seafood, so kosher fishes are acceptable. (Non-kosher shellfish and mollusks would also be acceptable for the Muslim.) In some versions of Shiite Islam, only kosher fishes are acceptable for Muslims and Shiite scholars specifically refer to the Jewish law in this regard. The argument is that what God has forbidden remains forbidden. However, many Shiites disregard this ruling and eat shellfish and mollusks as well.

Just to qualify my answer, I’m an orthodox Jew. No, Jews cannot eat Halal. As Muhammad Arafat Azam answered below, the slaughterer must be a Jew who is trained specifically. The knife is sharpened each time. There can only be a single stroke of the knife. If the knife gets caught or if the slaughterer pauses, the meat is not kosher. It is forbidden for us to eat blood. So, then the animal is hung upside down to drain the blood. Dirt is sprinkled on the blood and the blessing is recited. Obviously the methods vary somewhat between poultry and larger animals like cows and sheep. Depending on the kind of animal, further checks are required. The animals organs are inspected for injury or disease. A sore on the lungs can indicate that the animal would not have lived out for another year, so it’s not kosher. The meat is then applied with coarse salt and soaked to drain out additional blood. There are time limits on when all of these things must be done.

Just to clarify, this only applies to kosher meat. As I understand it, halal only applies to meat. Kosher is much more broad in the sense that it applies to everything a Jew eats. The term “Glatt Kosher” is what applies only to meat. If meat is kosher but not “glatt” kosher, then it means that some minor sores were found on the lungs. Some observant Jews will eat this.

Two of my employees are religious Muslim ladies. We were recently discussing this topic, and I found it very interesting. One of the ladies said her husband eats kosher, but she does not, and her children do not. She only eats Zabiha Halal. She told me that with regular halal, the slaughter can be done with a machine. An animal cannot be considered kosher if slaughtered with a machine.