I have heard that many Orthodox married couples never sleep in the same bed, even during the times that they are permitted to each other. Coming from a non-Orthodox background, this seems shocking to me, since I was brought up to believe that sleeping in the same bed is a major part of being married. Is there any source for this practice? Is it halacha?
People are generating a lot of unnecessary fog here, especially with the weird asceticism-within-marriage idea. Let's not go there, please.
A couple will need two separate beds for ~12--15 days per month when the wife is cycling normally; plus at least a few weeks after childbirth. Hence it's very common to have two beds that can be separated (or put together). If there's some other arrangement that works for them, that's fine too.
For the times of the month that the couple doesn't need to separate the beds, it's entirely their decision what to do about the beds. (There are plenty of non-Jewish couples who like having their own space.) It's not shocking that some may find it easier to just keep them separated then too -- and if the couple is happy with that, that's fine and good. There's nothing that can or should be dictated about this.
I found this "Halacha" in the Kaf HaChaim, Orach Chaim 240:63
ישכב תמיד במטה מיוחדת בפני עצמו. ואם צריך לשמש לשם מצות פו׳׳ר אחר גמר .השימוש כמו חצי שעה יקום וישוב למטה היוחד לו. אור הצדיקים סי׳ כז׳ אות ג׳
Furthermore, in Sefer Piskei Teshuvos, (pamphlet on Siman 240, footnote 226), the author quotes his father as being against the practice of having a single bed. (I have a sefer, entirely about Orach Chaim 240, and he quotes a sefer called דרכי טהרה. I have never heard of this sefer, nor can I find it, but apparently it has something relevant to this discussion.)
The Nitai Gavriel brings an opinion that the two beds should never be connected, or to have a double bed, due to Maris Ein. So from a practical perspective, sleeping in the same bed can be difficult, as it essentially means sleeping in one bed while leaving a second one empty, not connecting them, that is likely not all that large.
I know of a Rov who paskened about having beds beyond a certain size that it is not Tznius. The specific size isn't relevant for this answer because it would be about what is standard to have in homes.
And many people just prefer/get used to sleeping separately. So when you put it all together, it does add up to a lot of reasons why sleeping the same bed is just impractical and thus not done.
There is, however, no strictly Halachic prohibition against it. I'm confident asserting that because in that whole chapter the Nitai Gavriel (who is a rather thorough collector of stringent opinions) says nothing (that I could find).
איבעיא להו: נדה, מהו שתישן עם בעלה היא בבגדה והוא בבגדו? - אמר רב יוסף, תא שמע: העוף עולה עם הגבינה על השלחן ואינו נאכל, דברי בית שמאי. בית הלל אומר: לא עולה ולא נאכל. - שאני התם דליכא דיעות. הכי נמי מסתברא, דהיכא דאיכא דיעות שאני, דקתני סיפא, רבן שמעון בן גמליאל אומר: שני אכסניים אוכלין על שלחן אחד, זה אוכל בשר וזה אוכל גבינה - ואין חוששין. - ולאו אתמר עלה, אמר רב חנין בר אמי אמר שמואל: לא שנו אלא שאין מכירין זה את זה, אבל מכירין זה את זה - אסורים. והני נמי - מכירין זה את זה נינהו - הכי השתא התם - דיעות איכא, שינוי ליכא. הכא - איכא דיעות, ואיכא שינוי.
The Gemara Shabbos Daf 13 (above) Questions whether a husband may share a bed with his wife who is a Niddah if they are wearing garments. The Gemara goes on to explain the rationale behind the question (Why would it be permitted?), the reasoning is that wearing their clothing while sleeping together is a shinui (change from usual conduct) and is therefore comparable to eating milk and meat at the same table with a shinui which is permitted. Thus we see that it was considered the norm to sleep with ones wife without clothing. With clothing would be considered a Shinui. PLEASE NOTE: The Gemara finalizes that sleeping in the same bed even clothed is prohibited during Niddah.
Many Orthodox couples use twin beds vs. queen or king size beds. This is done for practical purposes. During the niddah period, as you have understood, the couple may not share the same bed. Thus, having twin beds makes the separation easier, otherwise, someone would probably have to sleep in a different room. As explained in the comments, the twin beds are pushed together, usually, during the non-niddah period, thus creating one "large bed".
AFAIK, there is no halachic prohibition, per se, to sleep in separate beds during the non-nidah week. However, I have seen in the Talmud (don't recall exact place, offhand) the rule that in general, men should refrain from excessive sex so that women not be turned into "chickens". Thus, it's possible that using separate beds (e.g. Fred & Wilma Flintstone style - with the night table between the beds) may be done as a way to discourage this excessive activity.
Despite the basic parameters of the laws of marriage that would basically allow married couples to sleep in the same bed at times, there are those who recommend that married couples keep and use separate beds at all (or nearly all) times. That is, to those that feel this way, there should be a permanent reminder in the bedroom of the importance of family purity.
From this perspective, this is both a matter of personal integrity as well as education for other members of the household (children, especially) and those who may enter the bedroom at times (visitors, cleaning help, etc.).
This is quite apart from the practical matter of needing two beds at certain times. That itself can necessitate a good deal of organizational planning from the interior design perspective - what do you do with two beds when they need to be separated vs. when they can be pushed together, and how do you fit bedroom furniture around this mobile arrangement. Again, my answer is besides this practical issue, there is the consideration of what is appropriate in the bedroom.
Source: Mesorah and guidance from my (non-Hasidic) rabbi when I was engaged.
In the Babylonian Talmud, the Kethuboth comprise the second tractate of the Order of Nashim, and deal with particular guidelines relating to married life in its various aspects and manifestations. That is, this particular tractate enumerates the privileges and duties of the husband and wife in their mutual relationship from the day of their betrothal.
Thus readers find in the Babylonian Talmud (b. Kethuboth 1:1, I.11.A [Folio 4a]) that the guidelines for married couples are to sleep one with another, except for those periods of mourning and/or menses.
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The Yerusalem Talmud provides additional detail concerning the expectation of husband and wife sharing the same marital bed. That following comes from Chapter 4 of the Kiddushin (קידושין, "Betrothal"), which deals with the initial stage of marriage and betrothal.
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This portion of the tract cites and comments on the relevant Mishnah source, and then provides guidance that suggests that husbands and wives share the same marital bed where physical contact ("flesh touching") may occur. Notwithstanding that they share the same marital bed, they may also cover themselves with their own bedcovers (and therefore be sleeping apart, but in the same bedroom).
Neusner, Jacob (2011). The Babylonian Talmud: A Translation and Commentary. Peabody, MA: Hendrickson Publishers, Kethuboth 1:1, I.11.A
Neusner, Jacob (2008). The Jerusalem Talmud: A Translation and Commentary. Peabody, Massachusetts: Hendrickson Publishers, Kiddushin 4:11, II.1.A.