The typical central monastery stupa was a massive hemispherical dome (figs. 1 and 2), called the anda, which, according to tradition, housed relics of the Buddha. 
This comes from the later Buddhist tradition found in the 5th-16th century CE Ashokavadana which holds that the Emperor Asoka distributed the actual relics of the Buddha in 84,000 stupas constructed all over India. This tradition is also found in the 5th century CE Sri Lankan Mahavamsa. For more details see John S. Strong, The Legend of King Ashoka: A Study and Translation of the Ashokavadana (Princeton: Princeton University Press, 1983) and John Strong, Relics of the Buddha (Princeton: Princeton University Press, 2004).
The dome was decorated at the top with three superimposed umbrellas, the chhatravali, and was surrounded by a small railing called the vedika. The whole structure was surrounded by a larger fence, also called a vedika, through which the devotee could enter via four toranas or gateways (fig. 2). The devotee could then circumambulate the stupa on a designated path called the pradaksinapatha. The toranas and vedika set the central stupa apart from the rest of the structures in the monastery complex. 
For further definitions and analysis of the stupa beyond this brief discussion, see Debala Mitra, Buddhist Monuments (Calcutta: Sree Saraswaty Press Ltd., 1971), pp. 8-56; Anna L. Dallapiccola and Stephanie Zingel-Avé Lallemant, The Stupa: Its Religious, Historical and Architectural Significance (Wiesbaden: Steiner, 1979); and the glossary in Kurt A. Behrendt, The Buddhist Architecture of Gandhara (Leiden: Brill, 2004), pp. 305-310.
Surrounding the great central stupa were both monastic residences and many smaller stupas. These smaller stupas housed the relics of other buddhas and well-known monks. These stupas were housed in shrines which in turn were located in larger courts. 
Here I follow the architectural classification scheme of Behrendt, The Buddhist Architecture of Gandhara, p. 309. He defines a stupa shrine as an "[a]rchitectural relic temple that houses a stupa." These shrines are located in largerstupacourts which Behrendt defines as "the part of the public sacred area made up of the main stupa and surrounding small stupas, relic and stupa shrines, and in the later periods, image shrines.".
These large complexes, located on the outskirts of urban centers, were places of worship for monks and the local lay community. 
In central India, the most famous of these massive hemispherical stupas are at Bharhut and Sanchi. In the northwest of the subcontinent, there are similar stupas at Taxila, Shah-ji-ki-dheri, Butkara, and Takhi-i-bahi, to name but a few.
At Taxila, while the Dharmarajika stupa and surrounding structures certainly fit the above description of a typical extensive monastery complex, within the city of Sirkap itself the stupa shrines were not connected to any central, massive stupa. Thus, rather than large, open spaces for worship, in the urban plan of Sirkap the stupas themselves were housed in shrines, often without a proper court. These stupa shrines had spatial relationships tied to the non-monastic structures around them rather than the monastic structures.
Figure 1: Line Drawing of Great Stupa at Butkara. Reference: Domenico Faccenna, Butkara I, Swat Pakistan, (1956–1962), Part I, IsMEO, ROME 1980 (image under GNU Free Documentation License).
Figure 2: Great Stupa at Sanci showing anda, vedikas, and toranas.Photo: Silver Gelatin Developing Out Paper (OGZ); Kern Institute, Leiden University (image under GNU Free Documentation License)